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Distributed Optimization of Sustainable Power Dispatch and FlexibleConsumer Loads for Resilient Power Grid Operations
by
Pirathayini Srikantha
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Graduate Department of Electrical & Computer EngineeringUniversity of Toronto
c© Copyright 2017 by Pirathayini Srikantha
Abstract
Distributed Optimization of Sustainable Power Dispatch and Flexible Consumer Loads for Resilient
Power Grid Operations
Pirathayini Srikantha
Doctor of Philosophy
Graduate Department of Electrical & Computer Engineering
University of Toronto
2017
Today’s electric grid is rapidly evolving to provision for heterogeneous system components (e.g. in-
termittent generation, electric vehicles, storage devices, etc.) while catering to diverse consumer power
demand patterns. In order to accommodate this changing landscape, the widespread integration of cyber
communication with physical components can be witnessed in all tenets of the modern power grid. This
ubiquitous connectivity provides an elevated level of awareness and decision-making ability to system
operators. Moreover, devices that were typically passive in the traditional grid are now ‘smarter’ as
these can respond to remote signals, learn about local conditions and even make their own actuation
decisions if necessary. These advantages can be leveraged to reap unprecedented long-term benefits that
include sustainable, efficient and economical power grid operations. Furthermore, challenges introduced
by emerging trends in the grid such as high penetration of distributed energy sources, rising power
demands, deregulations and cyber-security concerns due to vulnerabilities in standard communication
protocols can be overcome by tapping onto the active nature of modern power grid components.
In this thesis, distributed constructs in optimization and game theory are utilized to design the
seamless real-time integration of a large number of heterogeneous power components such as distributed
energy sources with highly fluctuating generation capacities and flexible power consumers with varying
demand patterns to achieve optimal operations across multiple levels of hierarchy in the power grid.
Specifically, advanced data acquisition, cloud analytics (such as prediction), control and storage sys-
tems are leveraged to promote sustainable and economical grid operations while ensuring that physical
network, generation and consumer comfort requirements are met. Moreover, privacy and security consid-
erations are incorporated into the core of the proposed designs and these serve to improve the resiliency
of the future smart grid. It is demonstrated both theoretically and practically that the techniques pro-
posed in this thesis are highly scalable and robust with superior convergence characteristics. These
distributed and decentralized algorithms allow individual actuating nodes to execute self-healing and
adaptive actions when exposed to changes in the grid so that the optimal operating state in the grid is
maintained consistently.
ii
Acknowledgements
I would like to sincerely thank my advisor, Professor Deepa Kundur, for her invaluable mentorship
and guidance. Her incredible patience, motivation and insights have allowed me to realize many of my
academic and career goals. She has continuously exposed me to many interesting research and teaching
opportunities that have been extremely rewarding. Being a doctoral candidate at the University of
Toronto has been such a revitalizing and wonderful experience for me mainly due to her. I am forever
indebted to her for her kindness and support.
I would also like to thank Professor Lacra Pavel, Professor Wei Yu and Professor Josh Taylor for
inspiring my work with their courses and research perspectives. I would also like to thank Professor
Hamed Mohsenian-Rad, Professor Reza Iravani, Professor Raymond Kwong, Professor Jorg Liebeherr
and Professor Ben Liang for participating in my thesis committees and providing me with constructive
feedback. Moreover, I am grateful for being able to engage in deep and illuminating discussions about
research, career and academia in general with members of the Communications and Energy Systems
groups.
Finally, I would like to acknowledge my parents and my brother for being such incredibly important
figures in my life. They have lifted me up from so many challenges. I am grateful for their unwavering
faith in me and for continuously bestowing upon me infinite love, support and strength.
“Life is momentary! Live it completely!” ∼ OSB
iii
Contents
1 Introduction 1
1.1 Traditional Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Smart Grid and Open Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Distributed Generation and Storage . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Flexible Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Energy Deregulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 Cyber-enablement and Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Problem Statement and Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 State-of-the-Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Demand Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Economic Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.3 Combined DR and DG dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Power Grid Optimizations 14
2.1 Convex Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Convex versus Non-convex Optimization Problems . . . . . . . . . . . . . . . . . . 16
2.2 Optimization for the Electric Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Power Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Steady-State Power Flow Problem Formulation . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Demand Response Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Economic Dispatch Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.5 Linear Power Flow Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.6 Voltage Regulation Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Demand Response 27
3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 DR Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.3 System Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Demand Response via Water-Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
iv
3.2.1 Demand Response Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.3 Dual of the DR Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.4 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.5 Theoretical Convergence Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.6 Analogy to Water-Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Demand Response via Population Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Set of Transformations to the DR Problem . . . . . . . . . . . . . . . . . . . . . . 39
3.3.2 Computation of EPU Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.3 A Game Theoretic Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.4 Resilience of EPU Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.5 Strategy Revisions by DR Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.6 Theoretical Convergence Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Distributed Generation Dispatch 53
4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1 Distributed Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.3 Dispatching DGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.4 System Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Distributed Dispatch via Dual Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Economic Dispatch Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.2 Master and Agent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.3 Termination Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.4 Summary of Proposed Dispatch Algorithm . . . . . . . . . . . . . . . . . . . . . . 61
4.2.5 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Distributed Dispatch via Population Games . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 Dispatch Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.2 Master Tier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.3 Secondary Tier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.4 Equivalence between Game Characterization and Optimization . . . . . . . . . . . 71
4.3.5 Theoretical Convergence Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.6 Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 Hierarchical Optimization of Distributed Generation and Demand 82
5.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2 Transmission Network Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
v
5.2.1 Introduction to ADMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.3 Transformations to ADMM Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.4 Summary of Decentralized Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.6 Comparison with State-of-the-art Decentralized Techniques . . . . . . . . . . . . . 95
5.3 Distribution Network Level Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.1 Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 Conclusions 97
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Appendix: List of Acronyms 101
Bibliography 102
vi
List of Tables
3.1 Distributed algorithm via water-filling for DR agent. . . . . . . . . . . . . . . . . . . . . . 35
3.2 Distributed algorithm via water-filling for the EPU. . . . . . . . . . . . . . . . . . . . . . 36
3.3 Distributed load curtailment via population game theory. . . . . . . . . . . . . . . . . . . 44
3.4 Comparison of DR methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1 Distributed dispatch algorithm via dual decomposition. . . . . . . . . . . . . . . . . . . . 61
4.2 Strategy revisions by DG agents with voltage rise considerations. . . . . . . . . . . . . . . 69
4.3 Voltage feasibility check by DG agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Comparison of dispatch methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.1 Decentralized algorithm for bus agent n in the transmission tier. . . . . . . . . . . . . . . 90
5.2 Comparison of results from central versus decentralized algorithms WECC 9 bus system. . 93
5.3 Comparison of results from central versus decentralized algorithms IEEE 39 bus system. . 94
5.4 Decentralized algorithm for bus agent n at the transmission tier. . . . . . . . . . . . . . . 96
vii
List of Figures
1.1 Illustration of a smart grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Diversity in generation mix in the smart grid. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Rendering of a smart city. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 NIST Framework representing active players in the smart grid. . . . . . . . . . . . . . . . 5
1.5 NIST Framework representing communication flows in smart grid. . . . . . . . . . . . . . 6
1.6 Interactions between chapters in the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Examples of convex and non-convex sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Examples of convex and non-convex functions. . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Simple electric grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Load profile of a home during winter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Power flow across a line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 Smart home energy management system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Sample load profile of a home. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Demonstration of water-filling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Aggregate demand in a neighbourhood containing 100 homes. . . . . . . . . . . . . . . . . 37
3.5 Reduced power operation for a single iteration. . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Regular versus reduced aggregate demands over a day. . . . . . . . . . . . . . . . . . . . . 38
3.7 Simplex and system potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Illustration of system dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.9 Pairwise comparison revisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.10 Performance of EGT based DR scheme under system limitations. . . . . . . . . . . . . . . 48
3.11 System subjected to perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.12 Ability of the EGT based DR algorithm to maintain demands around a setpoint. . . . . . 49
3.13 Comparison between PC protocol and SG method. . . . . . . . . . . . . . . . . . . . . . . 50
4.1 System model for economic dispatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Sample system model for DG dispatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Power dispatched by DGs for the three types of α computations. . . . . . . . . . . . . . . 62
4.4 Real-time dispatch over a day. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 Power Savings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6 System model for economic dispatch via master and secondary tiers. . . . . . . . . . . . . 66
4.7 Local feasibility checks with voltage rise considerations. . . . . . . . . . . . . . . . . . . . 68
4.8 Low-voltage distribution network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
viii
4.9 Topology of cyber-physical DN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.10 Projection revisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.11 Impact of population size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.12 Recovery of DG Agents from Disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.13 System behaviour (real power and bus voltages) without DG dispatch in place. . . . . . . 78
4.14 Real-time dispatch (real power and bus voltages) over a day with dispatch solution in place. 79
4.15 Comparison of Dispatch Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 System model for hierarchical topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 WECC 3 bus line diagram, adapted from reference [123]. . . . . . . . . . . . . . . . . . . . 91
5.3 IEEE 39 bus line diagram, adapted from reference [27]. . . . . . . . . . . . . . . . . . . . . 91
5.4 Residual for WECC 9 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 Residual for IEEE 39 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.6 Impact of Lagrangian step-size on ADMM convergence. . . . . . . . . . . . . . . . . . . . 93
ix
Chapter 1
Introduction
The electric power grid is a critical infrastructure that empowers a broad spectrum of crucial operations in
modern societies. Recent concerns regarding climate change and energy monopoly have spurred a rapidly
evolving landscape in the power grid. Thus, it is possible to witness the onset of remarkable changes in
fundamental power system components due to increasing diversity in generation mix, consumer demands,
grid protection mechanisms and regulatory processes [1].
1.1 Traditional Grid
For more than hundred years, grid operators have primarily strived to provision for guaranteed uninter-
rupted power supply to consumers [2]. Due to the widespread adoption of a vertically integrated power
grid model with stringent regulations in place, system operators have had incredible insights into various
nuances of the grid. This has enabled the implementation of well-defined planning and control processes
to manage day-to-day operations in the grid [3]. For instance, system operators can forecast consumer
demands using historical data fairly accurately since electricity consumption patterns are traditionally
strongly correlated with diurnal and seasonal changes. Moreover, as the generation mix in the grid has
been primarily composed of bulk generation entities such as fossil, hydro and nuclear plants; generation
capacities of these can be readily inferred from fuel inventories and physical laws (e.g. those associated
with hydro-electric generation). These measures are then leveraged by system operators to centrally
plan well in advance (e.g. hour or day-ahead) the amount of power that can be dispatched by every
active generator to economically meet consumer demands using simplified system models. This process
is referred to as economic dispatch.
Large synchronous power plants, located at remote locations for reasons that include safety (e.g.
nuclear plants) and ease of fuel access (e.g. gas, coal, hydro), produce power at pre-defined levels in
accordance to the centrally computed economic dispatch reference levels. This power is transported over
long distances via high-voltage transmission networks in order to reduce line losses and delivered to con-
sumers residing within distribution networks (DNs) at a lower voltage more suitable for end appliances.
Instantaneous fluctuations in demands that deviate from forecasts are typically accommodated for by
automatic generator control (AGC) built into large synchronous power sources [4]. AGC is a decentral-
ized control mechanism that infers imbalance in demand and supply through local frequency deviations
and adjusts generator power inputs to restore balance. In order to further ensure reliable power supply,
1
Chapter 1. Introduction 2
redundancies in generation and transmission are engineered into the design of the grid so that the sys-
tem will still be able to operate even when one major power component fails (e.g. transmission line or
generator) and this is referred to as N-1 contingency. Grid protection mechanisms such as relays and
circuit breakers are typically passive and are activated by sudden surges in local currents [5]. These
devices isolate incident locations so that issues resulting in instability do not propagate throughout the
grid.
This demand/supply and protection framework has been highly effective in managing the power grid
for many decades until very recently.
1.2 Smart Grid and Open Challenges
Due to mounting concerns regarding climate change and rising energy costs, achieving reliable power
grid operations is no longer the primary goal for system operators. A noticeable shift is now evident
in energy policies that now vigorously promote sustainable, economical and reliable power grid opera-
tions. This has resulted in efforts that strive to minimize dependence on power plants that are expensive
and associated with high carbon-footprint. Renewable distributed energy sources, storage systems and
inherent demand flexibility are considered to comprise of vastly untapped potential for facilitating sus-
tainable grid operations [1]. Energy deregulation has been embraced recently to encourage competitive
energy markets that can then serve to drive electricity prices down to lower rates and prevent energy
monopoly [6]. Moreover, cyber-enablement in the power grid has empowered many individual system
components with the ability to monitor, communicate and intelligently actuate [7]. These capabilities
allow for resilience to be built into the core of the system.
CONTROL
CONTROL
CONTROL
CONTROL
Figure 1.1: Illustration of a smart grid.
These upcoming and emerging trends summarized in Fig. 1.1 have resulted in a movement that has
transformed the traditional power grid into a smart grid. A formal definition of the smart grid presented
by the National Institute of Standards and Technology (NIST) is as follows [8]:
“Smart Grid develops and implements measurement science underpinning modernization of the Na-
tion’s electric grid in order to improve system efficiency, reliability and sustainability, by incorporating
distributed intelligence, bi-directional communications and power flows, and additional advancements to
Chapter 1. Introduction 3
create a smart grid.”
These changes bring upon many opportunities as well as many challenges due to significant deviations
in the traditional operational paradigm of the power grid and these are discussed next.
1.2.1 Distributed Generation and Storage
Power generation in the smart grid now consists of a diverse mix of energy sources as illustrated in
Fig. 1.2. Renewable generation sources combined with storage systems are highly preferred alternatives
Figure 1.2: Diversity in generation mix in the smart grid.
to traditional fossil-fuel based synchronous plants as these have zero emissions and are not typically
associated with transmission line losses due to the physical proximity of deployment to consumers [9].
Distributed generators (DGs) such as solar panels and wind turbines are typically deployed by individual
power consumers (e.g. rooftop panels and micro wind turbines) or in large dedicated wind/solar farms.
Storage systems are becoming widely commercialized in consumer markets with the advent of electric
vehicles [10]. With significant penetration of DGs that operate in tandem with storage systems, DNs will
be minimally dependent on unsustainable central power plants that are based on fossil-fuels (e.g. coal
and gas generation) to supplement local consumer demands and even have opportunities to make profits
by capitalizing on recent deregulations in energy markets to inject surplus generation back into the
main grid. However, a major deterrent in capitalizing on these opportunities is the inherent generation
variability of DGs.
Renewable DG generation is dependent on uncontrollable external environmental factors such as solar
irradiance (affected by cloud cover) and wind speeds (dependent on atmospheric pressure). Renewable
generation forecast models are associated with large error margins that tend to increase over longer
prediction horizons. For instance, day-ahead prediction models for wind generation are associated with
error margins that range between 20% to 40% whereas predictions over one minute intervals are very
accurate [11, 12]. Hence, day-ahead planning involving a large number of DGs will be an extremely
difficult task for system operators due to these perilously high error margins. For this reason, most DGs
are not dispatched by the system operators but are set to operate at maximum power point (MPP)
tracking levels so that all instantaneously generated power is injected back into the grid [13]. However,
with concentrated DG penetration especially in DNs, this maximum power injection approach is not
physically viable as the existing low-voltage infrastructure does not support bi-directional power flows.
Excessive reverse power flow can result in bus over-voltage conditions that can then lead to adverse
equipment damages [14]. Moreover, when power supply from DGs is not accurately accounted for in
day-ahead planning, reliably meeting consumer demands will be very expensive as significant number
Chapter 1. Introduction 4
of costly ancillary generation entities must be commissioned. Furthermore, balancing demand with
supply can even be infeasible when available generation capacities of standby operational reserves are
unable to offset imbalance stemming from unexpected drops in renewable generation. In these cases,
AGC mechanisms will not be sufficient in maintaining system balance within the prescribed stable
region. Furthermore, contingency planning will be very difficult and passive protection mechanisms such
as circuit breakers and relays will be tripped unnecessarily due to significant volatility in generation.
Recovery from these service interruptions can be very expensive and time-consuming.
1.2.2 Flexible Consumers
With the advent of the internet of things (IoT), it is possible to observe significant changes in consumer
demand patterns as illustrated in Fig. 1.3. With the recent commercialization of electric vehicles, storage
Figure 1.3: Rendering of a smart city.
systems and DGs for homes, consumer demand patterns can no longer be accurately predicted using
only diurnal and seasonal changes or historical data [15]. Unsustainable generation sources such as gas
plants are commissioned typically during peak consumer demand periods as these are able to ramp up
and down generation rapidly as needed to serve fluctuating demand peaks. These generation sources are
typically expensive and inefficient. During non-peak hours, base load demands are primarily served by
nuclear and hydro plants that operate at a constant level and produce no carbon emissions. However,
as these plants are inflexible and require significant time for ramping up/down generation levels, these
are not utilized to supplement highly fluctuating peak demands [16].
As discussed in the previous section, DGs can be utilized to reduce dependence on peak generation
plants. Another alternative will be to curtail consumer demands during these peak periods [17]. Cur-
tailing or shifting consumer demands via direct control or monetary incentives is referred to as demand
response (DR). Typically, direct load control (DLC) techniques are implemented in industrial and com-
mercial sectors. These sectors consist of large load entities that commit to various power reduction levels
well in advance. These are then activated by electric power utilities (EPUs) during times of need in
exchange for monetary incentives. DLC is not widely implemented in the residential sector. Instead, the
Chapter 1. Introduction 5
time-of-use (TOU) method is used in many regions such as Ontario where electricity prices are set well
in advance for various time periods over a day. Since TOU prices are higher during peak periods, con-
sumers are incentivized to use appliances during non-peak periods. It is however possible for consumers
to become desensitized to these price differentials. Moreover, TOU prices are not reflective of the actual
costs incurred in the power grid (e.g. real cost of generation, congestion) [18].
Most consumers have inherent flexibility in their demand patterns and the degree of this flexibility
is captured by the consumer comfort level measure in this thesis [19]. For instance, consumers may
be tolerant to their thermostat setpoints operating around particular thresholds. Appliances such as
washing machines, dishwashers or dryers can operate at a later time in a day. Moreover, if consumers own
storage systems, these can be used as buffers for storing electricity during inexpensive periods. This also
prevents the wastage of surplus power generated by DGs. Home energy management systems (HEMSs)
can be employed by the EPU to coordinate appliances in a home, based on individual consumer flexibility
and the state of the power grid [9]. However, this is not a straightforward task as there are thousands of
residential consumers with individual tolerance limits and preferences on how various appliances operate.
Coordinating all of these entities centrally is not a tractable task for the EPU. Hence, tapping onto the
significant potential of demand flexibility by system operators can be a daunting task.
1.2.3 Energy Deregulation
Energy deregulation has allowed entities such as private generation sources and investors who attempt
to take advantages of energy arbitrage to directly compete in electricity markets [20]. Modern grid
Figure 1.4: NIST Framework representing active players in the smart grid.
interactions of various traditional components with the energy markets, as conceptualized by NIST,
are illustrated in Fig. 1.4. The competitive structure of energy markets prevents energy monopoly
and serves to reduce electricity prices. However, many challenges regarding how ownership can be
assigned to non-revenue generating grid maintenance processes (e.g. transmission line expansion) and
how a large number of individual private entities can be coordinated at a system-wide level to ascertain
reliability are major concerns. Traditionally, demand and supply planning has been straightforward as
individual generation characteristics of synchronous plants were readily available to a central managing
authority. Moreover, infrastructure expansions required to support growing consumer demands are
typically managed by these central public entities as well. With deregulation, transmission lines can
operate close to physical limits and congestion can result in cascading system-wide failures. As it is
Chapter 1. Introduction 6
not possible to segregate power flowing through the transmission lines, assigning ownership for causing
congestion to various private generation entities is a difficult task. Moreover, committing public funds for
expensive infrastructure enhancement projects based on private generation entities that may not exist
in the future are not sound investment plans. Hence, the coordination of diverse individual participants
that takes into account physical infrastructure limits in addition to economical benefits is imperative for
the uninterrupted operation of the grid.
1.2.4 Cyber-enablement and Resilience
The widespread integration of communication technologies with the physical grid has resulted in an
information-rich cyber-physical system that consists of interconnected components equipped with the
ability to make intelligent decisions [21]. Various communication networks in the smart grid are illus-
trated in Fig. 1.5 which is a diagram rendered by NIST. For instance, phasor measurement units (PMUs)
Figure 1.5: NIST Framework representing communication flows in smart grid.
collect local grid measurements at a high frequency (around 50 Hz) and relay these to data concentrators
(DCs) that then apply analytics on this data for grid monitoring purposes. Homes are equipped with
smart meters that record energy consumption measurements at high granularity and communicate these
to the corresponding EPU. This data is then used by the EPU for billing purposes and/or to provide
consumer feedback for incentivizing sustainable energy consumption patterns [22]. Smart appliances
in homes such as thermostat controls, refrigerators, lights, washers, and dryers are equipped with the
ability to intelligently adopt local operations based on user preferences and behaviours [23]. Moreover,
critical power system components such as generation systems (both traditional synchronous and dis-
tributed generation), storage and grid protection components are also equipped with communication
and intelligent decision-making capabilities.
This elevated level of monitoring and actuation capabilities facilitated by the cyber-enabled power
grid can be leveraged to coordinate heterogeneous system components via advanced data analytics
and control techniques. Furthermore, adaptive decision-making by individual entities allows for the
automated recovery from any stress imposed due to unexpected circumstances and thereby enhancing
Chapter 1. Introduction 7
the resilience of the system. According to reference [24], a system is resilient if it is able “to prepare for
and adapt to changing conditions and withstand and recover rapidly from disruptions”. In this thesis,
resilience to perturbations arising from noise in communication signals (injection of false information) and
unresponsiveness of actuating nodes is considered. To alleviate adverse effects from these perturbations,
the system must be robust to perturbations whereby unaffected active nodes are able to actively adapt
and respond to disturbances in the system. This allows for automated recovery and in-built resilience
in the system.
However, it is important to note that the standard communication protocols utilized by cyber-enabled
devices also inherit well-known vulnerabilities [25]. These can be exploited by adversaries to gain access
to actuating components and stealthily manipulate these to cause major system disruptions [26, 27].
Thus, it is imperative to ensure that security mechanisms are incorporated into the core of smart grid
solutions so that system components can thwart adverse effects from these insidious attacks or natural
disasters [28].
1.3 Problem Statement and Thesis Objectives
The opportunities and challenges stemming from the smart grid paradigm present many rich and re-
warding research opportunities. As such, in this thesis, methods that enable the seamless plug-and-play
integration of a large number of heterogeneous power components such as consumers with flexible de-
mands and DGs to achieve sustainable, optimal and resilient grid operations at both the distribution
and transmission network levels are investigated. Demand response and economic dispatch are disparate
operations that are traditionally considered separately in a centralized manner. Typical challenges as-
sociated with centrally managing these entities stem from the lack of scalability, single point of failure
issues and sub-optimality. In this thesis, a distributed approach is utilized that enables the coordination
of these active nodes to achieve common goals of sustainability and economical grid operations in a
scalable and robust manner by leveraging on theoretical constructs from convex optimization and game
theory. Thus, following are the objectives of this thesis:
• Extend DR to a large number of residential consumers to coordinate flexible loads so that depen-
dence on peak unsustainable synchronous generation sources can be minimized while also account-
ing for consumer comfort levels.
• Propose economic dispatch for DGs deployed at large numbers in the DN level while accounting
for voltage rise limits in the buses.
• Propose real-time distributed techniques that optimally combines DR and DG dispatch over every
one minute interval to account for high variability in DG generation and demand fluctuations
without being subjected to large error margins from forecast models.
• Theoretically and practically demonstrate strong convergence properties of these proposed tech-
niques to ensure resilience against unexpected or adversarial disturbances.
• Combine DR and DG across multiple distribution networks into a single hierarchical scheme that
coordinates power flow across the transmission network level to minimize network congestion.
Chapter 1. Introduction 8
1.4 State-of-the-Art
State-of-the-art proposals in the existing literature in the context of DR and economic dispatch are
presented next. These proposals can be broadly classified into centralized, decentralized and distributed
categories that are either real-time (e.g. timescale is in minutes) or used in advanced planning (e.g.
timescale is in hours).
1.4.1 Demand Response
DR proposals represent adjustments made by the EPU to consumer power demand patterns via mecha-
nisms that include controlling thermostatically operating appliances around a threshold, shifting/curtailing
appliance usage and/or providing monetary incentives to influence consumer demand patterns.
Central DR techniques involve a consolidating entity usually a representative of the EPU that cen-
trally coordinates appliance operations (typically thermostatically controlled loads) in participating
residential homes. Central methods are advantageous as EPUs can exercise greater control over the
system. References [29–31] directly control heating, ventilating and air-conditioning (HVAC) systems
via stochastic optimization, queuing theory and simplified forecasting of appliance usage. The com-
putational complexity of solving these DR formulations grows exponentially as the size of system (i.e.
number of DR participants) increases. Hence, sophisticated resources with significant computational
power are required for the practical implementation of these techniques. Moreover, maintaining and
managing consumer preferences that are not consistent in a central repository will result in significant
communication overheads and storage requirements. Furthermore, these techniques do not scale well in
large systems consisting of different types of appliances (both thermostatically and non-thermostatically
controllable loads), as discrete operating points will render the DR formulation intractable. Thus, these
techniques cannot be applied for generalized DLC in the residential sector.
Decentralized and distributed techniques are designed to allow participating entities to capitalize
on the ubiquitous communication capability in the smart grid to iteratively arrive at an optimal DR
solution by exchanging information amongst each other. As a central entity does not directly manage
the operation of end nodes, single-point-of-failure risks are not applicable to these techniques. Moreover,
information about local operational conditions of end nodes is abstracted from the central entity. This
reduces concentrated information flow. Additionally, privacy is preserved as local information is not
exposed to an external entity. These techniques are prone to false data injection attacks in information
exchanges. The consequences entailed due to these are explored in reference [32] which investigates how
malicious false information propagation affects distributed schemes that primarily rely on peer-to-peer
information exchanges. Distributed DR algorithms proposed in the existing literature leverage upon
various theoretical constructs in agent-based optimization. Consensus-based algorithm in reference [33]
engages in DLC via iterative exchanges of local states amongst neighbours. Reference [34] utilizes the
sub-gradient method to arrive at the optimal solution in a distributed manner while taking into account
spatial and temporal appliance constraints. Reference [35] utilizes a hierarchical technique based on dual
decomposition to coordinate demand resources. Another proposal in reference [36] leverages congestion
pricing commonly used in the internet to manage traffic for adaptive electric vehicle charging based
on local preferences. Reference [19] entails direct actuation of inductive and resistive components of
appliances and therefore practicality of this proposal is questionable. Convergence rates to optimality of
the afore-mentioned techniques are dependent on the number of participants in the system. Larger the
Chapter 1. Introduction 9
system, the greater will be the time required for convergence. Reference [37] utilizes adaptive pricing
based on mathematical models for thermal and deferrable loads which may not accurately reflect the
actual operational patterns of these loads.
Game theoretic techniques are also widely applied in DR schemes. References [38–40] capitalize on the
HEMS present in smart homes to schedule appliances and storage systems in advance based on electricity
prices. These schemes require prior knowledge on what appliances will be operating at which time well
in advance (typically a day-ahead) and are not scalable for a large system with thousands of participants.
Moreover, not all appliances can participate as these schemes focus only on ‘shiftable’ appliances such as
dishwashers, dryers, etc. Reference [41] utilizes non-cooperative and evolutionary game theory to model
how residential users and multiple EPUs interact with one another. Reference [42] utilizes population
games for DR to account for individual consumer comfort budgets, however with no guarantees on the
uniqueness or robustness of the optimal solution. Hence, the ensuing system can be subject to instability
and ringing with minor perturbations.
1.4.2 Economic Dispatch
Next, state-of-the-art in the economic dispatch literature is presented. Demand and supply in today’s
grid is based on a wholesale electricity market that is administered by an independent system operator
(ISO) which seeks to balance consumer demands with available generation in participating energy sources
in an economical manner. Any generator injecting power into the transmission system must participate
in this wholesale electricity market and submit an offer containing details on the amount of electricity
it can supply and the associated price it will levy based on day-ahead demands forecasted and posted
publicly by the ISO. These generators may belong to a vertically integrated system or operate privately as
independent power plants (IPPs). Spinning reserves include generators and/or storage systems that are
operating in standby mode. These are expected to come online in the event of a disruption or unexpected
demand spikes. These ancillary services are paid “operating reserve” payments by the ISO [43]. These
bids can continue to arrive until two hours before the actual dispatch takes place. After this point, the
ISO ranks the submitted bids in the order of increasing price and accepts bids starting from the lowest
price until all of the forecasted demands are met. The market clearing price is uniformly set to the
price specified in the last accepted offer. Security constrained load flow studies are incorporated into
these auctions that are conducted hours in advance. Sensitivity analysis is also incorporated in real-time
dispatch across 5 minute intervals in order to account for inaccuracies in demand forecasts [44].
As mentioned in Sec. 1.2.1, DG sources are typically not dispatched and inject generated power
at maximum possible levels. As DG penetration grows significantly, this model of DG integration
presents great difficulties with the planning processes necessary for the effective operation of wholesale
electricity markets. Load flow analysis conducted hours in advance may not be accurate due to forecast
errors in DG power supply. More standby reserves that are typically expensive will be necessary to
accommodate unexpected changes in DG generation. Moreover, sensitivity analysis conducted in real-
time for transmission lines will not be sufficient to ascertain that all physical limits are heeded as DGs
are increasingly being deployed in low-voltage distribution networks. Existing proposals in the literature
that address these issues are discussed next.
Steady-state power flow constraints consist of sinusoidal components which are non-linear. With
heavy penetration of DGs, these constraints cannot be ignored. Centralized dispatch schemes proposed
in the literature attempt to tackle this difficulty by leveraging on prediction, stochastic optimization,
Chapter 1. Introduction 10
approximations and heuristics techniques. Moreover, security concerns due to single point of failure
are inevitable in centralized schemes. References [45–49] utilize stochastic, prediction and dynamic
programming methods to dispatch DGs at an advanced time period and this allows the ISO to utilize
ample computing resources to perform security assessment. However, due to the significant error margins
and variabilities present in DGs, a wide margin of risk must be incorporated into the dispatch techniques.
This will not allow the ISO to fully leverage the generation potential of DGs in an economical manner.
Other proposals in references [50–55] utilize heuristic techniques that are quicker to solve but do not
guarantee solution optimality and convergence speed. Hence, these can result in suboptimal solutions
that are more costly. Even if there is some tolerance to sub-optimality by the ISO, not using forecast
models will entail every DG transmitting local generation capacity constraints at a high frequency to
the ISO. This will result in significant storage and communication overheads. Reactive power dispatch
for voltage regulation is considered in reference [56]. References [57, 58] apply approximations such as
linearization and semi-definite programming (SDP) relaxations to non-linear constraints in the power
flow problem. Linearization techniques are typically used in day-ahead scheduling of generation systems.
Although SDP relaxations result in tighter bounds than linearization techniques, these may still result
in infeasible solutions. Applying these for real-time coordination of DGs can result in the infringement
of physical bus voltage and line limits. SDP relaxations can be exact in radial networks such as DNs
with very strict restrictions on system settings that can be impractical.
Distributed and decentralized techniques offload computational and communication intensive oper-
ations from the ISO to individual participating DG sources that are cyber-enabled and equipped with
intelligence. References [59–64] focus on balancing overall power demands with supply without taking
into consideration the underlying physical constraints. These utilize techniques such as sub-gradient
methods, distributed dynamic programming and consensus methods to achieve economic dispatch of
DGs. Convergence of these techniques is proportional to the number of participants in the dispatch
problem. Game theoretic approaches are used in references [65–70] for dispatch. Specifically refer-
ences [69, 70] take into account constraints with respect to power and voltage limits. Scalability is of
main concern with these proposals. Reference [71] proposes using only local state measurements and
no communications to make dispatch decisions that heed physical constraints. This technique does not
guarantee an optimal solution as the overall system is not taken into account in the dispatch. Refer-
ences [72–76] are decentralized techniques based on dual decomposition and alternative direction method
of multipliers (ADMM) which take into account physical grid constraints. References [72–74] apply con-
vex relaxations which are exact in distribution networks under stringent conditions. Reference [75]
applies ADMM to the original non-convex grid constraints with no guarantees on the optimality of the
solution. Furthermore, distributed techniques that depend on information exchanges amongst peers are
prone to false data injection attacks.
1.4.3 Combined DR and DG dispatch
So far, existing proposals in the areas of DR and DG dispatch have been considered separately. Proposals
that combine these two mechanisms to complement one another are presented next. Reference [77]
proposes a method that considers economic dispatch for a small to medium scale system while accounting
for both demand flexibility and physical network limitations. Reference [78] utilizes a primal-dual method
for load-side frequency control. These methods are tailored specifically for frequency control and cannot
be easily generalized to take into account other considerations such as voltage regulation. Reference [79]
Chapter 1. Introduction 11
proposes optimal frequency regulation by evoking DLC for generators and DR using a linearized power
system model.
1.5 Thesis Outline
In light of the challenges and limitations present in the existing literature with securely managing a large
number of highly fluctuating heterogeneous power components such as flexible consumer appliances and
DGs, this thesis presents a detailed investigation on how these difficulties can be effectively overcome and
proposes novel techniques that can be utilized to enable the seamless integration of these components
while accounting for physical grid constraints. More specifically, this thesis consists of six chapters. First,
a background on the theoretical tools leveraged in this thesis in the areas of convex optimization and game
theory is summarized in Chapter 2. Next, in Chapter 3, an overview of novel DR methods proposed for
coordinating a large number of flexible consumers while accounting for individual consumer preferences
is presented. These techniques are based on principles of water-filling in convex optimization, dual
decomposition and population game theory. Next in Chapter 4, two novel real-time economic dispatch
techniques for DGs are discussed. The first technique utilizes sub-gradient methods and does not account
for physical grid constraints. The second technique, which is based on dual decomposition and population
game theory, accounts for physical constraints. Chapter 5 presents a hierarchical algorithm that combines
DR and DG dispatch in distribution and transmission networks. This method capitalizes on ADMM,
decomposition and population games. Finally, the thesis is concluded in Chapter 6 and future directions
are also presented here. The general logical flows between the chapters are depicted pictorially in Fig.
1.6.
Figure 1.6: Interactions between chapters in the thesis.
The following presents a detailed description of the content in each chapter:
• Chapter 1: The premise of this thesis is presented along with a detailed overview of challenges
Chapter 1. Introduction 12
and opportunities present in today’s smart grid. A general overview of existing literature in the
areas of DR and DG dispatch is presented. The formal problem statement and contributions of
this thesis are outlined.
• Chapter 2: Main problem formulations commonly constructed and solved in the power grid by
system operators are presented in this chapter. Prior to this, a background on convex sets and
functions is presented which is useful for identifying the tractability of solving an optimization
problem. Then, the optimal power flow problem is introduced that consists of a general objec-
tive which takes into account net power flow balance across buses, bus voltage limits, generation
capacity constraints and flexible demand curtailment ranges. Challenges associated with this for-
mulation are highlighted. Then, various flavours of this formulation such as the DR problem,
economic dispatch problem, linearized power flow problem and voltage regulation problem are pre-
sented. Analysis of various challenges associated with these formulations with respect to convexity
is presented.
• Chapter 3: The DR problem which takes into account discrete load curtailment levels and com-
fort budgets of thousands of residential consumers is addressed in this chapter. First, a distributed
solution based on water-filling in convex optimization is applied to a simplified version of the orig-
inal DR problem formulation without taking into account discrete load curtailment levels. This
technique allows for a simplistic introduction to the distributed solution structure commonly inte-
grated into all proposals presented in this thesis. Then, another proposal that actually takes into
account the discrete load curtailment levels is presented which is based on decomposition and pop-
ulation game theoretic constructs. Theoretical convergence results along with practical simulation
studies using realistic models are presented. The resilience of the solution to perturbations is also
theoretically and practically demonstrated.
• Chapter 4: Dispatch of a large number of DG sources is investigated in this chapter. First,
general economic dispatch is considered which does not take into account the underlying physical
constraints of the grid. A distributed solution based on dual decomposition and the sub-gradient
method is presented to coordinate a large number of highly fluctuating DGs in an optimal manner.
Flux in generation capacities of DGs are accounted for by considering very short optimization
intervals. Next, another dispatch solution based on decomposition and population game theory
is presented that takes into account the underlying physical grid constraints to accommodate a
large number of DGs in the DN. As low-voltage networks such as the DN are sensitive to excessive
power injections, this proposal which takes into account voltage rise considerations is practical
for these systems. Comprehensive practical and theoretical results on various characteristics of
these proposals that include convergence rates, resilience to perturbations and scalability are also
presented.
• Chapter 5: In this chapter, a hierarchical technique is proposed that combines DR and DG
dispatch in multiple DNs situated across the transmission network. This heterogeneous solution
exacts coordination across two tiers. The transmission tier involves completely decentralized co-
ordination amongst buses using ADMM and consensus to achieve optimal linear power flow. This
abstracts details about individual systems such as IPPs participating in energy markets, DGs in
the DN, flexible loads and bulk generation entities from the system operator. Then, at the DN
Chapter 1. Introduction 13
level, flexible consumers and DGs are coordinated so that the setpoint computed in the transmis-
sion network level can be achieved. This proposal is also accompanied by theoretical and practical
verifications.
• Chapter 6: The thesis is concluded in this chapter. A brief summary is presented and the
contributions of the thesis are identified. Possible future extensions of this work are also presented.
Chapter 2
Power Grid Optimizations
The process of curtailing power demands and dispatching DGs in order to achieve a specific system
objective (e.g. minimizing costs, carbon footprint, etc.) while ensuring that the physical grid and
consumer constraints are met, constitute the formulation and solving of an optimization problem. An
optimization problem is composed of optimization variables, objective(s) and constraints. In the problem
formulations presented in this thesis, associated optimization variables include loads curtailed by flexible
consumers and power injected by DGs into the grid. The grid operator seeks to minimize the system
cost resulting from various configurations of the optimization variables and this is accounted for by the
objective(s) of the problem formulation. The optimization variables are subject to constraints stemming
from underlying physical system properties and/or resource limitations. All points that satisfy these
constraints form a feasible set. This set is composed of various values that the optimization variables
can take without infringing upon any of the imposed constraints. An assorted set of problems that include
optimal load curtailment, economic dispatch, linear power flow and voltage regulation can be formulated
using different combinations of system objectives and physical constraints as detailed later in this chapter.
Certain problem formulations are more suited than the others under particular circumstances. However,
before discussing these, a brief exposition on the notion of convex sets and functions is presented next as
these are imperative in gauging the difficulty of solving an optimization problem. Pertinent information
about the problem formulation that includes scalability and resource requirements can be obtained from
this assessment.
2.1 Convex Sets and Functions
An optimization problem can be classified as either a convex or a non-convex optimization problem. Con-
vex optimization problems can be solved using commercially available solvers [80]. These are tractable
problems and the computational complexity of solving these is typically polynomial with respect to the
size of the problem (i.e. number of optimization variables). On the other hand, non-convex optimization
problems are intractable and the complexity of solving these increases exponentially with the size of
the problem. Hence, non-convex problems are not scalable and impose significant challenges in a large
system. Thus, identifying whether a problem is convex or not provides vital information about the
difficulty entailed in solving the problem. All theoretical constructs listed in this section are based on
the material presented in reference [81].
14
Chapter 2. Power Grid Optimizations 15
2.1.1 Convex Sets
Constraints in an optimization problem define a set of feasible values that can be taken by the opti-
mization variables. Any point not contained within this set will result in the infringement of one or
more constraint(s). A convex optimization problem must have a convex feasible set. The definition of a
convex set C is as follows:
θx+ (1− θ)y ∈ C ∀ x, y ∈ C, θ ∈ [0, 1] (2.1)
Intuitively, this translates to the following. If all points on lines drawn between any randomly selected
points x and y from the set C lies within the same set, then C is a convex set. A set is non-convex if the
definition in Eq. 2.1 does not hold. Simple two-dimensional examples of convex and non-convex sets are
presented in Fig. 2.1.
Figure 2.1: Examples of convex and non-convex sets.
The set depicted in the upper left hand corner is formed by the intersection of five linear inequality
constraints. According to the intuitive description of a convex set presented in the above, it is evident
that this set is indeed convex (lines connecting any two randomly selected points from the polygon lie
within that polygon). The second set located in the upper right hand corner constitutes of points on
a curve. This set is formed by a nonlinear equality constraint and an example of this is a quadratic
equality constraint. It is clear that the points on the curve do not form a convex set as it is possible to
identify more than one line formed by connecting two points on this curve that consist of points lying
outside of this set. Next, in the bottom left hand corner, three discrete points form the set. Evoking the
intuition behind convex sets, it is evident that this is also not a convex set. The final set located in the
bottom right hand corner is formed by a linear equality constraint and all the points on this line form
a convex set. These results can also be derived theoretically by evoking the formal definition presented
in Eq. 2.1 as detailed in reference [81].
2.1.2 Convex Functions
In the objective of an optimization problem, functions are typically used to assign costs to various values
taken by the optimization variables. Like sets, functions f : Rn → R can also be classified as convex or
non-convex. The formal definition of a convex function is listed in Eq. 2.2.
f(θx+ (1− θ)y) ≤ θf(x) + (1− θ)f(y) ∀ x, y ∈ dom(f), θ ∈ [0, 1] (2.2)
Chapter 2. Power Grid Optimizations 16
This definition of a convex function can also be intuitively interpreted as follows. All points on a line
drawn between any two points on the function must lie above that function within the domain range
specified by the initially selected two points. Fig. 2.2 presents two examples of functions. Evoking this
intuitive definition, it is clear that the quadratic function on the left of the figure is a convex function
while the sinusoidal function on the right is not convex.
Figure 2.2: Examples of convex and non-convex functions.
2.1.3 Convex versus Non-convex Optimization Problems
An optimization problem is composed of objectives (i.e. cost functions) and constraints (i.e. feasible
set(s)). Each constraint constitutes of a set of points and the intersection of all constraint sets in the
problem formulation results in the feasible set. These are points that do not infringe upon any of the
restrictions/limits set in the problem. If any one of the constraints represents a non-convex set, then the
entire feasible set results in being non-convex as well. For instance, optimization problems with discrete
variables are typically known to be difficult to solve [82]. Moreover, if the cost function in the objective
is not convex, then the optimization problem is classified as a non-convex optimization problem. The
difficulty of solving a non-convex problem is typically identified as NP-hard [81,83]. Hence, in a convex
optimization problem, all constraint sets and cost functions must be convex. This notion of convexity is
used to classify the difficulty of solving optimization problems specifically in the context of the electric
grid in the next section.
2.2 Optimization for the Electric Grid
The electric grid consists of a large number of diverse components. Today’s grid primarily consists of
alternating current (AC) power components. Hence, grid states take values in the complex space (i.e.
consists of real and imaginary parts). In this thesis, all symbols representing various complex attributes
of the grid will be listed in capital letters. Vectors will be represented by non-indexed symbols.
2.2.1 Power Components
For the sake of brevity, main components in the power grid that include generators, loads, buses and
lines along with associated constraints and costs are discussed next. Fig. 2.3 presents the single line
Chapter 2. Power Grid Optimizations 17
Figure 2.3: Simple electric grid.
diagram of a very simple grid for illustrative purposes1. All system configurations considered in this
thesis are based on balanced three-phase circuits.
Generators
Generators output real and reactive power as required by the grid and these include traditional syn-
chronous power plants, DG sources and storage systems. These convert energy from a specific domain
(i.e. hydro, chemical, mechanical, etc.) into electricity to meet consumer power demands. Due to the
AC nature of the grid, there are two types of power flowing in the system and these are real and reactive
power. Real power denoted by the symbol p (measured in Watts (W)) is used to perform actual work
whereas reactive power q (measured in volt-amperes reactive (VAR)) is present when complex voltages
and currents in the AC system are not in phase with one another. Complex power S measured in
volt-amperes (VA) is a combination of real and reactive power: S = p + iq where i =√−1. Apparent
power also measured in VA is the magnitude of the complex power |S|. In Fig. 2.3, there are two power
sources. The generator attached to the bus labelled 3 is associated with a power rating of 800 MVA
and injects 520 MW of real power into the system. Every power source is associated with individual
generation capacity requirements that pose limits on the upper and lower bounds of real and reactive
power generation as follows:
cpg ≤ pg ≤ cpg ∀ g ∈ G (2.3)
cqg ≤ qg ≤ cqg ∀ g ∈ G (2.4)
where G is the set of all generators in the system, g is the index assigned to a generator in the set G, cpg and
cqg represent the lowest possible real and reactive power that can be generated by generator g, cpg and cqg
are the associated upper limits on real and reactive power and pg and qg are the actual real and reactive
power generated by generator g. Traditional power plants are associated with ramping constraints which
are not considered here as this thesis focuses on the dispatch of DGs with negligible ramp up/down
rates [84]. Additional constraints must be taken into consideration when sources generating discrete
levels of power are included in the generation mix. For instance, storage systems at homes may supply
power at finite number of discrete power ratings [10]. In these cases, additional constraints will be
1Image courtesy of reference [3]
Chapter 2. Power Grid Optimizations 18
imposed on pg as follows:
pg ∈Mg = {M1 . . .Mng} ∀ g ∈ Gd (2.5)
where the set Gd ⊆ G represents all generators with discrete power inputs and the setMg consists of ng
real power levels at which generator g can inject power into the grid. It is important to note that Eq.
2.5 represents a discrete set and therefore is not convex.
Loads
Loads represent entities belonging to the residential, commercial and industrial sectors that consume
power. The real power demand of consumer d is denoted as pd. If the consumer is using inductive loads
then reactive power demands must be taken into consideration as well and this is denoted as qd. Real
power curtailment by a flexible consumer d is denoted as prd. In Fig. 2.3, a set of loads are attached to
Bus 2 and consume 280 MVAR of reactive power and 800 MW of real power. Flexible consumers have
some tolerance to adjustments made to their real power demands. The power reduction by a flexible
consumer d can range between 0 to cd as follows:
0 ≤ prd ≤ cd ∀ d ∈ D (2.6)
where D represents the set of all consumers, the power reduction prd possible for consumer d can range
between 0 (i.e. no curtailment) to cd which is an upper bound on power curtailment imposed by local
preferences and appliance operating conditions which are discussed in Chapter 3. If the consumer is not
flexible then cd can be set to 0 so that the demand cannot be curtailed without infringing upon Eq.
2.6. Typically appliances in the residential sector operate at discrete power demand levels [18]. Fig. 2.4
illustrates the load profile of a home during the winter over a 24 hour period and the discrete nature of
appliance operations is evident in this plot.
0 5 10 15 20 250
500
1000
1500
2000
2500
3000
3500
4000Load Demand Snapshot of a Single Class 1 Home (Winter)
Time (hour)
Po
we
r (W
att
s)
Figure 2.4: Load profile of a home during winter.
Chapter 2. Power Grid Optimizations 19
Hence, various values that can be taken by real power reduction for each flexible consumer in the
residential sector is further constrained as follows:
prd ∈Md = {M1 . . .Mnd} ∀ d ∈ D (2.7)
where the setMd consists of nd real power levels at which consumer d can curtail local power demands.
Again, it is imperative to note that Eq. 2.7 represents a discrete set and therefore is non-convex.
Next, every flexible consumer has a particular tolerance for the curtailment of individual local loads.
This tolerance is referred to as consumer comfort budget and is captured by imposing a constraint that
ensures that the cumulative demand reductions over a day is well within the energy budget set in advance
by the consumer as follows:
E(prd) ≤ Edbudget ∀ d ∈ D (2.8)
where the function E(.) is an accumulation of all demand reductions so far over a day including prd and
Edbudget is the overall energy reduction budget over a day configured to be tolerated by consumer d.
In addition to individual constraints imposed on demand curtailment by each consumer, global
coupling constraints that ensure balance between overall real and reactive power demands and supply
are as follows: ∑g∈G
pg − plgrid =∑d∈D
pd (2.9)
∑g∈G
qg − qlgrid =∑d∈D
qd (2.10)
where the first term in the above two constraints represents overall real/reactive power generation, the
second term depicts all real/reactive power losses in the grid encountered during transmission and the
third term is the aggregate real/reactive power demand of all consumers in the system. As highlighted
in the upcoming sections, transmission losses are neglected in some formulations and included in others.
Certain systems are sensitive to bus voltages which are affected by the real and reactive power flows in
the power network. Computation of these values is discussed next.
Buses
Buses serve as connection points in the grid for various power components. All buses in the system are
represented by the set B. There are three types of buses in general and the classification of buses into
these three groups is generally based on two power system measures [87]. One is the complex power
injection Sb into bus b and the other is the complex bus voltage Vb. A slack bus is used as a reference
point for computations. The complex voltage associated with the slack bus is fixed whereas the complex
power injection can vary. A load bus, also referred to as the PQ-bus, has fixed complex power injection
with voltage that can vary. A generator bus b, also referred to as a PV bus, has fixed real power injection
(i.e. pg =Re(Sg)) and voltage magnitude (i.e. |Vg|), however, the reactive power qg and voltage angle
∠Vg can vary. In Fig. 2.3, bus 1 is connected to a power source and bus 2 is connected to loads only.
Bus 3 is connected to both generation and loads. The net power injection into bus 3 is therefore total
Chapter 2. Power Grid Optimizations 20
generation minus total demand in that bus as captured by the following set of relations:∑g∈Gb
pg −∑d∈Db
pd = Re(Sb) (2.11)
∑g∈Gb
qg −∑d∈Db
qd = Im(Sb) (2.12)
where Gb is a set representing all generators connected to bus b and Db represents all demands attached
to bus b. Buses with varying voltages are subject to the following limits to avoid irreversible equipment
damages:
|Vmin|2 ≤ VbV ∗b ≤ |Vmax|2 ∀ b ∈ B (2.13)
Bus voltage magnitudes are constrained to operate between the thresholds |Vmin| and |Vmax|. As Eq.
2.13 represents a set of quadratic inequalities, these represent convex constraints. Transformers serve
as interfaces between generation/loads to the transmission networks. As power is transmitted at high
voltages over long distances, the voltages are typically stepped up from generator buses and stepped down
at load buses. Constraints associated with the transformers are neglected in this thesis as transmission
and distribution networks are considered separately.
Lines
Lines connect various buses together in the power grid. Set of lines present in the power network that
is under consideration is denoted as L and a line connecting bus a ∈ B and b ∈ B is represented as
a↔ b ∈ L. Various interconnections in the power grid can be succinctly represented as a graph in which
each bus is a node and every line is an edge. Power losses in lines depend on the characteristics of the
corresponding lines and these are captured by line admittances. Hence, the magnitude of power pa,b
flowing across line a↔ b from buses a to b is not the same as that of pb,a which represents power flowing
in the reverse direction. The power flowing across a line at steady-state can be computed by evoking
Kirchhoff’s voltage and current laws as illustrated in Fig. 2.5.
Figure 2.5: Power flow across a line.
First, an assumption about the direction of power flow is necessary. In Fig. 2.5, it is assumed that
power is flowing from bus a to bus b. From the perspective of bus a, the power flow on a line connecting
to bus a via bus b is:
VaV∗a Y∗a,b − VaV ∗b Y ∗a,b
and from the perspective of bus b, the power flow on the line connecting to bus b via bus a is:
VbV∗b Y∗b,a − VbV ∗a Y ∗b,a
Chapter 2. Power Grid Optimizations 21
Hence, it is evident that the complex power flows Sa,b 6= Sb,a. Suppose that bus a serves as a connection
point for multiple lines, then the net power injected into the bus must be equal to the net power flow
across that bus via lines coinciding with it (i.e. ∀ a↔ b ∈ L ):
pa + iqa =∑
a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b)
Separating the real and imaginary parts result in the following set of relations for real and reactive power
injections into bus a in terms of bus voltages and admittances:∑g∈Ga
pg −∑d∈Da
pd = Re( ∑a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b)
)∀ a ∈ B (2.14)
∑g∈Ga
qg −∑d∈Da
qd = Im( ∑a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b)
)∀ a ∈ B (2.15)
The above net power balance equations are quadratic equality constraints in the complex domain which
are not convex. These represent constraints imposed by the underlying physics of the system which are
important especially in low-voltage DN networks that are sensitive to reverse power flows.
Cost Functions
There are numerous objectives that are typically taken into consideration by a grid operator. This
thesis focuses on DG dispatch and DR and therefore the optimization variables constitute of pg ∀ g ∈ Gand pd ∀ d ∈ D. Hence, the system costs incurred will be functions of these optimization variables.
Costs in the power grid are typically designed to be quadratic functions of the associated optimization
variables [88,89].
C(p) =∑g∈G
fg(pg) +∑d∈D
fd(pd) (2.16)
where p is a vector representing the optimization variables, C : R|D|+|G| → R and f : R → R. Every
participating entity i is associated with an individual cost fi. As these costs are summed over individual
participants in the system (generators and flexible consumers) to obtain the final cost, C(.) is referred
to as a separable function. Examples of C(.) in economic dispatch include pTAp where p = [p1 . . . p|G|]
is the vector representing the optimization variables and A ∈ R|G|x|G| is a positive semi-definite matrix
(i.e. A � 0). For separability, A will be a diagonal matrix. Intuitively, this translates to greater costs
for larger levels of dispatch.
2.2.2 Steady-State Power Flow Problem Formulation
So far, a brief introduction to the main power components in the grid and associated properties has
been presented. Next, a generalized formulation referred to as the optimal power flow (OPF) problem
composed of all of these components is introduced in POPF . This formulation is a steady-state prob-
lem. Transients in the system are considered to be absorbed by highly inertial bulk generation entities
commonly present in the grid. In stand-alone microgrid systems (not considered in this thesis), these
perturbations will not be negligible.
Chapter 2. Power Grid Optimizations 22
POPF : minp
C(p) =∑g∈G
fg(pg) +∑d∈D
fd(pd)
s.t.
cpg ≤ pg ≤ cpg ∀ g ∈ G (C1)OPF
cqg ≤ qg ≤ cqg ∀ g ∈ G (C2)OPF
pg ∈Mg = {M1 . . .Mng} ∀ g ∈ Gd (C3)OPF
0 ≤ prd ≤ cd ∀ d ∈ D (C4)OPF
prd ∈Md = {M1 . . .Mnd} ∀ d ∈ D (C5)OPF
E(prd) ≤ Edbudget ∀ d ∈ D (C6)OPF∑g∈G
pg − plgrid =∑d∈D
pd (C7)OPF∑g∈G
qg − qlgrid =∑d∈D
qd (C8)OPF
|Vmin|2 ≤ VaV ∗a ≤ |Vmax|2 ∀ a ∈ B (C9)OPF∑g∈Ga
pg −∑d∈Da
pd = Re( ∑a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b)
)∀ a ∈ B (C10)OPF∑
g∈Ga
qg −∑d∈Da
qd = Im( ∑a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b)
)∀ a ∈ B (C11)OPF
where Gd is a set representing all generators taking discrete generation values. As both DR and economic
dispatch are considered in the OPF, optimization variables of POPF are represented by the vector
p consisting of pd ∀ d ∈ D and pg ∀ g ∈ G. (C1)OPF to (C3)OPF are constraints associated with
the generators. Constraints (C4)OPF to (C6)OPF embody the consumer demand constraints. Global
coupling constraints balancing overall power generated with demands are captured in (C7)OPF and
(C8)OPF . Limits on bus voltages are imposed by constraint (C9)OPF . Net power balance at every
bus introduced by the underlying physics of the system is reflected in (C10)OPF and (C11)OPF . Even
though constraints (C7)OPF and (C8)OPF are implied in (C10)OPF and (C11)OPF , these are included
for ease of formulating certain problems that are presented later in this chapter. Constraints (C1)OPF to
(C11)OPF serve to capture resource limitations and physical grid characteristics. All of these constraints
have already been introduced earlier in Sec. 2.2.1.
Solving this OPF directly will be extremely challenging for the following reasons. First, parameters
corresponding to DG generation capacities and consumer demands (i.e. cpg, cpg,c
dg,c
dg) will be at con-
stant flux. Typically in the literature, forecast models are utilized to fix these parameters for a long
period of time. As these models are associated with high error margins especially over longer prediction
horizons, the optimization interval is reduced to one minute periods in this thesis. Then, much more
accurate forecast models can be leveraged by individual participants (flexible consumers and/or DGs)
to identify these parameters based on local conditions. This, however, imposes significant computation
and communication overheads which are overcome later in this thesis by leveraging upon distributed
and decentralized solution architectures. The second issue is introduced by the presence of non-convex
constraints in POPF . Constraints (C3)OPF and (C5)OPF represent discrete demand and generation
Chapter 2. Power Grid Optimizations 23
levels. In Sec. 2.1.1, discrete sets have been identified to be non-convex. Other non-convex constraints
in POPF are (C10)OPF and (C11)OPF which are associated with net power flow balance at each bus.
These are complex quadratic equality constraints. When the real and imaginary components are consid-
ered separately, quadratic and sinusoidal terms are present. As mentioned in Sec. 2.1.3, an optimization
problem becomes intractable even when there is one non-convex constraint. In this particular case, there
are |Gd|+ |D|+2|B| such constraints and this is proportional to the total number of nodes in the system.
Hence, directly solving this NP-hard optimization problem over every one minute interval for a large
system is certain to be infeasible.
To resolve these issues of practicality, in this thesis, the OPF is divided into simpler sub-problems
that are suitable representations of the system objectives and limitations under certain circumstances.
Underlying structures of these sub-problems are capitalized to propose more tractable and practical
solutions. Then, these sub-problems are combined into a hierarchical algorithm to coordinate DGs and
flexible consumers in a manner that heeds the constraints listed in POPF while also achieving minimal
costs for the system operator. As such, in the following, the four sub-problems considered later in this
thesis are introduced.
2.2.3 Demand Response Formulation
The DR problem listed in PDR, consists of flexible consumers whose demands are directly adjusted by the
EPU according to individual local operating conditions and comfort budgets so that the overall demand
reduction amounts to SDR at minimal cost. This setpoint SDR is determined by system operator and can
be designed to maintain peak demands around a particular value so that dependence on expensive and
unsustainable peak generating plants is minimized. The EPU provides some compensation to consumers
for participating in the DR program and this is captured in the objective of PDR. The total number
of constraints in this problem is 3|D| + 1. Moreover, many parameters in these constraints such as cd,
Md and Edbudget can vary based on individual consumer preferences. Gathering these parameters from
consumers at a fairly regular basis to accurately capture fluctuations in these will result in significant
communication overheads for the EPU. Moreover, constraint (C2)DR is not convex. Hence, with a large
number of participants, as typical in the residential sector, directly solving PDR becomes intractable.
PDR : minp
CDR(p) =∑d∈D
fd(prd)
s.t.
0 ≤ prd ≤ cd ∀ d ∈ D (C1)DR
prd ∈Md = {M1 . . .Mnd} ∀ d ∈ D (C2)DR
E(prd) ≤ Edbudget ∀ d ∈ D (C3)DR
SDR =∑d∈D
prd (C4)DR
However, problem PDR has a decomposable structure. The objective is a summation of costs incurred
by individual consumers and all constraints with the exception of (C4)DR can be separated into groups
that correspond to every participating consumer. This structure is leveraged in Chapter 3 to design
Chapter 2. Power Grid Optimizations 24
distributed DR.
2.2.4 Economic Dispatch Formulation
Economic dispatch (ED) problem listed in PED is used by an ISO to balance overall generation with
aggregate demands in a manner that is least expensive. Losses in the transmission lines, bus voltage
limits and net power balance constraints are neglected in this formulation.
PED : minp
C(p) =∑g∈G
fg(pg)
s.t.
cpg ≤ pg ≤ cpg ∀ g ∈ G (C1)ED
pg ∈Mg = {M1 . . .Mng} ∀ g ∈ Gd (C2)ED∑
g∈Gpg −
∑d∈D
pd = 0 (C3)ED
Certain generation sources are more costly than others and these are accounted for in the objective where
individual costs are summed across all participating generation sources. As local generation capacities
can vary significantly for DGs, parameters such as cg, cg and Mg must be obtained from individual
DGs when PED is solved centrally. This can be onerous especially when there is a large number of
active DGs whose generation parameters must be collected frequently in order to effectively capture
fluctuations in generation. Moreover, constraint (C2)ED is not convex. If every DG can intelligently
make its own decision regarding how much power to dispatch based on local operating conditions (i.e.
generation capacity) and general external information conveyed by the ISO, then computational and
communication overheads can be offloaded from the central dispatching entity. This is demonstrated in
Chapter 4 of this thesis by capitalizing on the separable nature of the problem PED.
2.2.5 Linear Power Flow Formulation
The main source of complexity in the OPF formulation in POPF is the set of net power balance equa-
tions that are quadratic equality constraints in the complex domain which are non-convex. In order to
overcome this difficulty, several approximations are applied to these equations so that these result in a
set of linear equations which greatly simplify computations as listed in constraint (C3)FR of PFR. This
linearized model is used for many applications in the power grid.
PFR : minp,θ
C(p, θ)
s.t.
cpg ≤ pg ≤ cpg ∀ g ∈ G (C1)FR
pg ∈M = {M1 . . .Mng} ∀ g ∈ Gd (C2)FR∑
g∈Ga
pg −∑d∈Da
pd =∑
a↔b∈L
−ba,b(θa − θb) ∀ a ∈ B (C3)FR
Chapter 2. Power Grid Optimizations 25
The optimization variables of PFR consist of the real power pg generated by each generator g and the
bus voltage angle θb at each bus b ∈ B. This reduction in the optimization variables is due to certain
simplifying approximations applied to the power balance equations in POPF as discussed next. Real and
complex power flow across line a↔ b ∈ L is repeated in the following for convenience:
pa,b + iqa,b = Sa,b = VaV∗a Y∗a,b − VaV ∗b Y ∗a,b
This can be equivalently expressed in terms of real terms as follows:
pa,b = |Va|2ga,b − |Va||Vb|ga,bcos(θa − θb)− |Va||Vb|ba,bsin(θa − θb)
qa,b = −|Va|2ba,b + |Va||Vb|ba,bcos(θa − θb)− |Va||Vb|ga,bsin(θa − θb)
where Ya,b = ga,b + iba,b and θ is associated with the complex voltage |V |∠θ. As outlined in [90], the
following assumptions are applied to the above set of equations:
1. |ga,b| << |ba,b|
2. sin(θa − θb) ≈ θa − θb; cos(θa − θb) ≈ 1
3. |Va| ≈ 1, |Vb| ≈ 1 (in per unit system) ∀ a, b ∈ B
which result in the simplification of the power balance equations to:
pa,b = −ba,b(θa − θb)
qa,b = 0
It is important to note that reactive power flows are eliminated by these simplifications. These assump-
tions are somewhat valid in high-voltage networks operating at normal conditions and can be applied to
simplify analysis. These, however, do not apply to low-voltage networks as the above assumptions will
not remain valid and result in bus voltages violating the specified nominal ranges.
2.2.6 Voltage Regulation Formulation
Voltage rise across buses in low-voltage networks is of significant concern for EPUs especially with the
recent proliferation of a large number of DGs in the DNs. DNs are designed to carry power from the EPU
substation to the consumers. It is not designed to handle excessive power flowing back into the grid. One
solution for rectifying this is to perform expensive infrastructure upgrades and allow DGs to operate at
MPP. Another alternative will be to consider bus voltage limits, reactive power and net power balance
to optimally integrate a large number of DGs into the DN by dispatching these as required. The voltage
regulation problem listed in PV R is very similar to the OPF problem. In this problem, the optimization
variables consist of only real power dispatched by DGs (flexible loads are not included). PV R is a more
comprehensive formulation than PED as net power balance at each bus along with voltage limits are
incorporated into the problem formulation.
Chapter 2. Power Grid Optimizations 26
PV R : minp
C(p) =∑g∈G
fg(pg)
s.t.
cpg ≤ pg ≤ cpg ∀ g ∈ G (C1)V R
cqg ≤ qg ≤ cqg ∀ g ∈ G (C2)V R
pg ∈M = {M1 . . .Mng} ∀ g ∈ Gd (C3)V R∑g∈G
pg =∑d∈D
pd −∑
a↔b∈L
pa,bl (C4)V R∑g∈G
qg =∑d∈D
qd −∑
a↔b∈L
qa,bl (C5)V R
|Vmin|2 ≤ VaV ∗a ≤ |Vmax|2 ∀ a ∈ B (C6)V R∑g∈Ga
pg −∑d∈Da
pd = Re( ∑a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b)
)∀ a ∈ B (C7)V R∑
g∈Ga
qg −∑d∈Da
qd = Im( ∑a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b)
)∀ a ∈ B (C8)V R
where pa,bl and qa,bl are real and reactive line losses encountered in line a↔ b ∈ L. Even though this for-
mulation is more comprehensive than PED, there are many more challenges accompanying this problem.
Discrete constraints in (C3)V R and non-convex power balance constraints in (C7)V R and (C8)V R con-
tribute to these difficulties. In the literature, convex relaxations are applied to these constraints which
are exact in radial networks such as the DN under certain stringent conditions. As these conditions
can be restrictive, convex relaxations are not considered for low-voltage DN networks in this thesis. A
distributed solution is introduced in Chapter 4 to overcome these challenges where PV R is divided into
master and secondary problems.
2.3 Remarks
In the above discussion, four problem formulations common in the power grid are introduced which are
suitable for various contexts in the power grid. These are presented in the order of increasing complexity.
In the culmination of this thesis in Chapter 5, a hierarchical distributed solution is presented that
combines various types of networks and problem formulations in a manner that is both practical and
scalable by capitalizing on the techniques introduced in Chapters 3 and 4.
Chapter 3
Demand Response
DR and DG dispatch are each considered separately in this chapter and the following chapter where
various approaches for handling these difficult problems are explored. These are then combined into a
single hierarchical solution in Chapter 5.
In particular, in this chapter, two distributed DR proposals are presented. The first proposal uti-
lizes the water-filling method in convex optimization and the second proposal leverages on theoretical
constructs from population game theory. These proposals are novel departures from the state-of-the-art
as these iterative solutions are designed to meet optimality and scalability goals in real-time (i.e. in the
scale of minutes). The main difference between the two proposals is the manner in which consumer de-
mand curtailment is modelled. In the first proposal, an assumption regarding consumer demand patterns
is made which significantly simplifies the original DR formulation in PDR whereas no such simplifying
assumptions are made in the second proposal, thus rendering it more generally applicable.
3.1 System Model
Details on the distributed and iterative approach utilized to design these two proposals are first presented
in this section. Then, various active components in the model along with assumptions made about the
system settings are introduced.
3.1.1 DR Structure
In a DR program, active participants are flexible consumers whose demands are adjusted by the EPU so
that a particular system goal can be met. The type of demand adjustment considered in this thesis is full
or partial curtailment of flexible loads. Partial curtailment can be achieved in loads consisting of flexible
power-consuming components such as heating elements (e.g. dryers operating at lower heat, heating
water at a lower temperature and so on). These consumers are associated with various comfort budgets
and are compensated for any load curtailment incurred while participating in the DR program. The
general structure of the two DR proposals presented in this chapter is distributed in nature. The EPU
broadcasts common signals to all participating entities at every signalling iteration. Flexible consumers
are each equipped with a HEMS which uses these signals to make appropriate decisions regarding
demand curtailment based on local operating conditions. As decisions are made by participating nodes
rather than a central entity, this solution architecture is considered to be distributed. Each HEMS is a
27
Chapter 3. Demand Response 28
representative of the EPU (like the smart meter) which acts on behalf of the EPU. Fig. 3.1 presents an
illustration of this DR setup.
Figure 3.1: Smart home energy management system.
The EPU capitalizes upon current aggregate grid measurements, which are readily available via
data concentrators, to compute signals that are broadcasted periodically. As these signals are general
and common to all active participants, the EPU will not need to initiate point-to-point communication
links with every active participant and transmit customized signals. This significantly reduces the
communication overhead for the EPU otherwise incurred due to bi-directional communication flows.
These lightweight signals containing a small set of numerical values are broadcast every 0.6 seconds
via the wireless medium. Even though the wireless medium is of broadcast nature, it is possible that
these consist of multiple hops which can result in excessive congestion when thousands of nodes are
exchanging information with the EPU. In our strategies, this is not the case as the EPU broadcasts the
signals in the downlink and capitalizes on the aggregate data made available in data concentrators. As
latencies entailed in transmissions across the wireless channel over a residential neighbourhood are in
the order of milliseconds, this signalling interval is large enough to account for communication delays in
uni-directional flows [21].
HEMS present in each active participant’s residence is equipped with a set of wireless receivers,
transmitters and an intelligent module. In this module, home owner d can program local preferences
regarding how appliances operate and these are translated into the cd parameter which serves as an upper
bound for local power demand curtailment in PDR. The comfort budget Edbudget parameter embodies
the total energy reduction due to demand curtailments that can be tolerated by the consumer over a
day and this can also be configured into the HEMS. The HEMS will be able to actuate local appliances
in the home via one of two methods. The first method leverages upon communication capabilities
in smart appliances to make any actuation decisions. For appliances that are not equipped with any
communication capabilities, smart plugs are used instead to control the operation of the appliance [91].
These plugs can communicate wirelessly and can be turned on or off remotely. The HEMS will be aware
of the current operating statuses of various appliances in the home and will compute actuation signals
using the signals broadcast by the EPU, local consumer preferences and operating conditions.
Chapter 3. Demand Response 29
The HEMS abstracts local operating conditions from the EPU and the EPU is concerned only about
the aggregate behaviour of the system. This decoupling significantly reduces communication and compu-
tational overheads for the following reasons. First, local operating conditions of thousands of consumers
are not transmitted to a central entity. HEMSs residing locally within consumer premises are responsible
for these local constraints. The EPU utilizes real-time aggregate measurements to compute signals that
convey information about the current state of the grid. This information is readily available in data
concentrators that are commonly deployed in DNs [1]. Each HEMS reacts to these signals and makes
appropriate local actuation decisions based on simple computations. Hence, majority of the concen-
trated computations are offloaded to thousands of individual HEMS devices. This distributed approach
parallelizes computations and takes advantage of the small-scale computational resources available in
HEMSs that are deployed in vast numbers throughout the system. The two proposals presented in this
chapter embody two different algorithms for computing these signals and local actuation decisions. In
both of these algorithms, the main design requirement is the guaranteed convergence of the iterative and
distributed decisions made by active participants to the optimal solution of the constructed DR problem
within the allocated optimization interval.
3.1.2 Assumptions
The assumptions made in the system model are summarized as follows:
1. EPU has access to current aggregate demand measurements via data concentrators;
2. EPU is fitted with a wireless transmitter that is used to periodically broadcast a signal containing
numerical values every 0.6s to DR participants;
3. Every flexible consumer participating in DR is equipped with an HEMS;
4. HEMS, also referred to as a DR agent, represents the EPU;
5. HEMS is equipped with intelligence and wireless communication capabilities to facilitate actuation
of local appliances based on consumer preferences and the reception of EPU signals;
6. The optimization interval set by the EPU is one minute in length and therefore demand and
generation in the system is considered to be constant over these one minute periods;
7. A large number of DR participants are present in the system;
8. There exists sufficient overall demands in the system that can satisfy the load curtailment goals of
the EPU; and
9. Compensation provided is a quadratic function with respect to the demand curtailment exacted.
Assumptions 1 to 5 align with the cyber-physical vision of the power grid as detailed in Chapter 1.
Assumption 2 is practical as this is taking place at a less granular scale in today’s grid whereby many
consumers are equipped with smart meters that can exchange information with the EPU at granularity
of minutes. Assumption 3 is now common in building management architectures where lighting and
temperature management technologies are managed via these EMS systems. Assumption 6 is necessary
to ensure that fluctuations in demand parameters are captured at a high granularity. Moreover, very
small error margins result from demand/supply forecast models applied over short time intervals such
Chapter 3. Demand Response 30
as one minute periods [43]. Assumption 7 naturally applies as DR proposals presented in this chapter
target the residential sector which is composed of thousands of participants. As the goal of the EPU is
to curtail some portion of existing demands in the system, assumption 8 is practical. Assumption 9 (also
applied in the existing literature such as reference [88]) offers greater compensation for higher levels of
demand curtailment.
3.1.3 System Model Implementation
In order to ascertain whether the proposed DR algorithms apply in practical systems, realistic models
are implemented in MATLAB to capture various nuances of the system accurately. Household demands
are highly correlated with many factors that include diurnal patterns, seasons, appliance penetration
rates and appliance usage patterns. There are |D| consumers participating in the DR program. Each
consumer represents a residential home. The load profile of each home is generated using parameters
provided in references [18, 92] that include penetration rates, power consumption patterns of common
home appliances and probability of active appliance usage based on the time-of-day and season in various
regions such as Ontario, France, India and Quebec. The appliances considered are washing machine,
dryer, dishwasher, electric stove, oven, freezer, fridge, water heater, space heater and air conditioner. All
load profiles generated in this chapter are based on parameters provided in reference [18] for Ontario. Fig.
3.2 illustrates a sample load profile generated for a home over a 24-hour period using these parameters
in the winter.
0 5 10 15 20 250
500
1000
1500
2000
2500
3000
3500Load Profile of a Home on a Winter Day
Time (hours)
Aggre
gate
Pow
er
(W)
Figure 3.2: Sample load profile of a home.
The periodic load signatures occurring throughout the day can stem from inductive appliances such
as fridge, freezer and air conditioner. Sudden bursts of power consumption occurring in the morning and
afternoon can result from resistive elements used by stoves which have an on/off power demand pattern
over short time intervals or other appliances such as dishwashers, dryers and washing machines.
The consumer can set preferences regarding the operation of various appliances. Demand curtailment
is not limited to completely switching off appliances. Appliances can also operate in conservation mode
where the invocation of resistive heating elements that consume significant amounts of real power during
operation is reduced [18]. For instance, a washing machine can operate using cold water cycles. A dryer
Chapter 3. Demand Response 31
can operate in tumble drying mode. A dishwasher can operate using only cold water. An air conditioner
can operate at a higher setpoint (on-cycles are less frequent) or not operate at all until the comfort budget
is exhausted. On the other hand, when the consumer does not want certain appliances to operate at
reduced modes, the power available for reduction in the home will be adjusted so that demands from
these inflexible loads are not altered. Hence, individual operation parameters of appliances in a home
are lumped into general parameters such as cd by the corresponding HEMS and this prevents details
from detracting from the main problem at hand.
In the numerical studies presented later for each DR proposal, the comfort budget Ebudgetd is set to
1 kWh for each participating flexible consumer. This is equivalent to the continuous reduction of power
consumption by 1 kW over an hour.
3.2 Demand Response via Water-Filling
In this section, the first DR proposal is presented which is based on the work published in reference [93]
by the author of this thesis. Water-filling technique in convex optimization is leveraged to achieve
distributed DR. First, the DR problem formulation (a slight variant of PDR) is presented, after which an
overview of the proposed algorithm is provided. Then, insights from theoretical and numerical studies
of the proposed algorithm are discussed.
3.2.1 Demand Response Formulation
The DR formulation used in the design of the proposed algorithm is listed in PpDR.
PpDR : minpr
C(pr) =1
2(pr)TApr
s.t.∑d∈D
prd = SDR (C1)WF
0 ≤ prd ≤ cd = min(pd, bd) ∀ d ∈ D (C2)WF
where pr is a vector of the form pr = [pr1 . . . pr|D|]
T , A ∈ R|D|×|D| is a diagonal matrix consisting of
strictly positive entries, pd is the current power demand of consumer d and bd is the maximum power
reduction possible for the current interval with respect to Ebudgetd and E(prd) of the consumer from the
previous optimization interval. Although this formulation is similar to PDR presented in Sec. 2.2.3, it is
important to note that there are no constraints discretizing the power reduction levels in PpDR. Hence,
the power reduction by each consumer is a continuous variable. The objective in PpDR is a strictly
convex function as A � 0. Moreover, as (C1)WF is a linear equality constraint and (C2)WF is a linear
inequality constraint consisting of constants, these form a convex feasibility set. Thus, PpDR is a convex
optimization problem. Although, not considering discrete power reduction levels can limit practicality,
this work provides deep insights into the design of real-time DR as discussed later. Moreover, if storage
systems with continuous power output levels are available in homes, then this formulation will be exact
as cd will translate to the state-of-charge (SOC) of the battery. In the second proposal presented later
in this thesis, discrete power reduction levels are considered and therefore is more generally applicable.
As PpDR is a convex optimization problem, off-the-shelf commercial solvers can be used to solve this
problem directly. However, extensive communication overheads will be incurred as parameters related to
the operation of various local appliances and preferences must be exchanged with a central coordinating
Chapter 3. Demand Response 32
entity. In order to avoid these difficulties, a distributed approach is used where DR agents (i.e. HEMSs)
representing the EPU reside at local premises of consumers and make iterative curtailment decisions
based on general information transmitted by the EPU every 0.6 seconds.
3.2.2 Proposed Algorithm
Here, a detailed overview of the proposed algorithm that is derived from PpDR is presented. There are
two important considerations in the proposed algorithm. First is the design of the signals transmitted
by the EPU at every signalling iteration and the second is the manner in which DR agents react to
these signals based on local operating conditions. To facilitate these processes, first the dual of PpDR is
constructed. Then, the feasibility conditions of the primal and dual problems are derived which are then
used to design the EPU signals and actuation actions taken by the DR agents.
3.2.3 Dual of the DR Problem
PpDR is referred to as the primal problem. As PpDR is a strict convex optimization problem, it has a
unique globally optimal solution [81]. The dual problem of PpDR is PdDR:
PdDR : maxλ1,λ2,ν
minpr
L(pr, λ1, λ2, ν)
where L(pr, λ1, λ2, ν) =1
2prTApr − λT1 pr + λT2 (pr − c) + ν(1T pr − SDR)
s.t. λd1 ≥ 0, λd2 ≥ 0 ∀ d ∈ D
where c = [c1 . . . c|D|]T , L(pr, λ1, λ2, ν) is the Lagrangian of PpDR and the dual variables are λ1 ∈ R|D|+ ,
λ2 ∈ R|D|+ and ν ∈ R. All constraints in the primal problem PpDR are moved into the Lagrangian. λ1
is a vector of Lagrangian multipliers associated with the lower bound of 0 imposed on prd in constraint
(C2)WF . λ2 is a vector of Lagrangian multipliers that correspond to the upper bound of cd applied to
prd in constraint (C2)WF . ν is the Lagrangian multiplier associated with the coupling equality constraint
(C1)WF in PpDR.
The Lagrangian multipliers λ1, λ2, ν are referred to as the dual variables. These dual variables can
be interpreted as penalties resulting when primal constraints (C1)WF and (C2)WF are violated. For
all feasible pr in PpDR, the following relation between the primal and dual objectives holds: C(pr) ≥L(pr, λ1, λ2, ν). The optimal value d∗ of PdDR coincides with the optimal value p∗ of PpDR only when
strong duality is satisfied [81]. Strong duality results when the relative interior of the constraint set
formed by (C1)WF and (C2)WF is not empty. Due to assumption 8 listed in Sec. 3.1.2, the Slater’s
condition holds for the system model considered in this proposal. When p∗ = d∗, there is no gap between
the dual and primal solutions. In this case, the dual problem can be solved instead of the primal problem
to obtain p∗.
3.2.4 Optimality Conditions
Optimality criteria for the primal and dual problems are divided into four categories and these are:
primal feasibility, dual feasibility, complementary slackness and stationarity. These form the Karush-
Chapter 3. Demand Response 33
Kuhn-Tucker (KKT) conditions for PpDR and PdDR as follows:
Primal feasibility:
prd∗ ≥ 0, prd
∗ ≤ cd ∀ d ∈ D
1T pr∗ = SDR
Dual feasibility:
λd∗1 ≥ 0, λd∗2 ≥ 0 ∀ d ∈ D
Complementary Slackness:
λd∗1 prd∗ = 0, λd∗2 (prd
∗ − cd) = 0 ∀ d ∈ D
Stationarity:
OprL(pr∗, λ∗1, λ∗2, ν∗) = 0
∂L
∂prd= addp
r∗d − λ1∗
d + λ2∗d + ν∗ = 0 ∀ d ∈ D
where add is the dth diagonal component of the cost matrix A in the objective of the primal problem and
pr∗, λ∗1, λ∗2, ν∗ are optimal primal and dual variables that result in the optimal values p∗ and d∗. Primal
feasibility conditions require that the optimal solution pr∗ satisfies constraints (C1)WF and (C2)WF of
the primal problem PpDR. Dual feasibility conditions require that the optimal solution λ∗1 and λ∗2 satisfy
the feasibility constraints of PdDR. Complementary slackness conditions eliminate the terms in the
Lagrangian that are associated with the primal inequality constraints so that the optimal solution of the
dual problem will be the same as that in the primal problem. Suppose that the ith inequality constraint
is generically represented as fi(pr) ≤ 0. If pr∗ takes value in the interior of the feasible set then the
associated Lagrangian multiplier λi of that constraint is 0. Otherwise, the constraints become activated
and fi(pr∗) = 0. Stationarity conditions, stemming from first-order optimality, require the gradient
of the Lagrangian with respect to the primal variables to be equal to 0. As the Slater’s condition is
satisfied in this system model, these KKT conditions are both necessary and sufficient. Hence, the KKT
conditions can be used to derive the optimal solution pr∗.
First, the complementary slackness KKT conditions are used to divide the values taken by λd1 and
λd2 into the following four groups:
1)λd1 = 0, λd2 = 0, then 0 < prd < cd
2)λd1 > 0, λd2 = 0, then prd = 0
3)λd1 = 0, λd2 > 0, then prd = cd
4)λd1 > 0, λd2 > 0, then prd is undefined
These groups established for λd1 and λd2 are substituted into the stationarity conditions in order to
eliminate the dual variables associated with the inequality constraints. This allows the specification of
Chapter 3. Demand Response 34
the power reduction variables prd only in terms of the variable ν as follows:
prd =
0, if ν ≥ 0,
−ν/add, if − addcd < ν < 0
cd, if ν ≤ −addcd
(3.1)
As ν is associated with the coupling constraint (C1)WF , optimal ν∗ will result when∑ni=1 p
rd = SDR.
Hence, the signals transmitted by the EPU are iterative updates made to ν based on the current offset
in∑ni=1 p
rd − SDR. Updates to ν are continually applied and broadcast by the EPU until the offset
becomes 0 at which point ν is inferred to be optimal. To facilitate the computation of ν, the EPU
acquires current aggregate consumer demand information from data concentrators (i.e. assumption 1).
Upon receiving these signals, every DR agent will revise local demands according to Eq. 3.1.
3.2.5 Theoretical Convergence Properties
Consider the scenario where every DR agent initiates local power reduction to prd = 0 at the beginning of
an optimization interval and then continues to increase reductions until the global coupling constraint is
satisfied. Suppose a function S(ν) =∑d∈Dmin(−ν/add, cd)−SDR is defined. The first term represents
the aggregation of the power reduction decisions made by each DR agent. It is evident that this function
S(.) is increasing with respect to ν. This implies that a unique solution for ν exists.
3.2.6 Analogy to Water-Filling
This method of revising local power reduction to achieve a global balance is akin to the water-filling
method typically utilized for optimal power allocation in multiple-input-multiple-output (MIMO) com-
munication systems [94]. The main difference lies in the orientation of the enclosure representing the
local constraints. Figure 3.3 illustrates how the water-filling analogy applies to the proposed distributed
method.
Figure 3.3: Demonstration of water-filling.
Chapter 3. Demand Response 35
For the purposes of illustration, suppose that add is fixed to 1. This results in a positive definite
matrix which implies that the objective of the primal problem is a strictly convex cost function. If −ν is
steadily increased from 0 and prd is set to min{−ν, cd} in accordance to Eq. 3.1, then plotting the power
reduction values for each consumer at a particular time instant will result in a bar graph depicted in Fig.
3.3. In this water-filling analogy, individual response of DR agents is similar to water levels resulting from
injecting water into an enclosure defined by individual demand curtailment constraints. It is evident
in this illustration that DR agents continue to increase power reductions until either maximum power
reduction levels are reached or until −ν stops changing. −ν reaches equilibrium when∑d∈D p
rd = SDR.
Distributed DR for DR Agent i
Receive signal ν from the EPU:cd ← min(pd, bd)if ν > 0 then
prd ← 0else if 0 > ν > −aiici then
prd ← −ν/addelse
prd ← cdend if
Table 3.1: Distributed algorithm via water-filling for DR agent.
Based on the signal ν received from the EPU at each signalling iteration, each DR agent will respond
according to Table 3.1. The EPU computes and transmits ν to all active DR participants in the system
using aggregate information about the system at every signalling iteration. There are many methods by
which the EPU can compute ν. One option will be to increase −ν steadily from 0 until the global coupling
constraint is satisfied. This information will be made available to the EPU by the data concentrators
(assumption 1). A more efficient method for the EPU will be to execute a binary search algorithm to
find the optimal value of ν by identifying the degree of imbalance in the global coupling constraint. For
instance, if during the current signalling iteration, the overall power reduction in the system is greater
than the setpoint SDR, then ν should be increased and otherwise decreased. Binary search that leverages
on these insights is detailed in Table 3.2. This search algorithm allows the EPU to systematically identify
the optimal ν∗ in an iterative manner whereby the search space is divided into half at every signalling
iteration [94].
In Table. 3.2, the search window for ν is defined by two variables νmax and νmin which represent
the maximum and minimum values ν can take. Initially, νmax is set to 0 and νmin is initialized to
−SDR/mind∈D
(add) as this is the lowest value ν can take to satisfy∑d∈D p
rd = SDR. Then, ν is set to be
the midpoint of νmin and νmax. Each signalling iteration t occurs every δ = 0.6 seconds (in accordance
to assumption 2), during which the EPU measures the deficit or surplus in demand curtailment (i.e.∑d∈D p
rd − SDR). Either νmin or νmax is updated to reduce the search space of ν into half. This is
repeated until the difference between νmin and νmax is below a threshold ε which is set to a very small
value.
Chapter 3. Demand Response 36
Distributed DR for EPU
t← 0a← min
d∈D(add)
νtmin ← −SDR/a, νtmax ← 0, νt ← (νmax −νmin)/2while abs(νtmax − νtmin) ≥ ε do
Broadcast νt to all DR agentsWait for δ secondst← t+ 1if∑d∈D p
rd − SDR > 0 then
νtmin ← νt
elseνtmax ← νt
end ifνt = (νtmax − νtmin)/2 + νtmin
end while
Table 3.2: Distributed algorithm via water-filling for the EPU.
3.2.7 Numerical Results
In order to illustrate the performance of the proposed DR algorithm, numerical studies are conducted
using the residential load models and appliance usage parameters outlined in Sec. 3.1.3. The simulated
system consists of 100 homes in a residential neighbourhood in Ontario over the winter season. Fig. 3.4
illustrates a snapshot of aggregate real power demands from these 100 homes over a 24-hour period. Two
periods of demand peaks are evident in this figure as typical for the winter season. The optimization
interval is set to one minute. Every interval is referred to as a DR cycle. EPU signalling takes place
every δ = 0.6s according to assumption 2 and this translates to 100 signalling iterations per minute. This
period allocated for signalling is practical as the communication capacity of a Zigbee wireless module
is in the order of kilobits per second (kbps) [21]. As each signal transmitted by the EPU consists of a
single value ν, these are lightweight signals which are subjected to latencies in the order of milliseconds.
Hence, this signalling period is practical.
In the first result presented in Fig. 3.5, the proposed DR algorithm is applied over one randomly
selected optimization interval (i.e. DR cycle) and the resulting aggregate demands in the system with and
without the distributed DR program in place are illustrated. According to assumption 6, demands are
considered to be constant over one minute intervals. For the optimization interval under consideration
in Fig. 3.5, the aggregate demand in the system is 10 kW when the proposed DR algorithm is not in
place. The proposed DR algorithm is used by the EPU to maintain aggregate demands in the system
around 9 kW. For this, SDR is set to 1kW. The resulting aggregate reduced demand is shown to converge
smoothly and rapidly to the targetted 9 kW setpoint with the proposed DR algorithm in place. More
specifically, only 8 EPU signalling iterations were necessary in this particular case for all 100 homes with
a heterogeneous mix of appliances to converge to the desired peak demand reduction level in the system.
In the next result depicted in Fig. 3.6a, the DR algorithm is applied over the entire day. In this
case study, the EPU maintains aggregate demands in the system around 20 kW. The dotted line in Fig.
3.6a represents aggregate demands with no DR in place and the red curve represents aggregate demands
Chapter 3. Demand Response 37
0 5 10 15 200
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
4 Aggregate Demand over a Winter Day
Time (hours)
Aggre
gate
Po
wer
(W)
Figure 3.4: Aggregate demand in a neighbourhood containing 100 homes.
0 20 40 60 80 1000
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Aggregate Power for a Single Cycle
Iterations
Ag
gre
ga
te P
ow
er
(W)
Reduced Operation
Regular Operation
Figure 3.5: Reduced power operation for a single iteration.
Chapter 3. Demand Response 38
0 5 10 15 200
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
4
Time (hours)
Aggre
gate
Pow
er
(W)
Aggregate Demand over a Winter Day
Reduced Operation
Regular Operation
(a) Aggregate demand setpoint is 20 kW.
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
4 Aggregate Demand over a Winter Day
Time (hours)
Aggre
gate
Pow
er
(W)
Reduced Operation
Regular Operation
(b) Aggregate demand setpoint is 15 kW.
Figure 3.6: Regular versus reduced aggregate demands over a day.
when the proposed DR algorithm is in place. It is clear that the EPU is able to maintain the aggregate
demands so that these do not exceed 20 kW.
If the EPU aims to reduce aggregate demands to a much lower setpoint such as 15 kW, then the
comfort budgets will become exhausted much more rapidly as illustrated in Fig. 3.6b. When comfort
budgets are exhausted rapidly, the EPU will not be able to maintain aggregate demands around 15 kW
throughout the day and individual consumer demands will eventually return to normal consumption
levels. Thus, demand curtailment cannot be sustained over a long period of time in this case. Fig. 3.6b
illustrates this behaviour as the aggregate demand is maintained around the setpoint from 5 a.m. to 11
a.m. After this point, due to the exhaustion of comfort budgets of certain consumers, aggregate demands
start to surpass this 15 kW setpoint and then the system eventually returns to the typical demand levels
over the remainder of the day. This example highlights the delicate balance between consumer comfort
tolerance levels and the goals set by the EPU. The EPU must be mindful of consumer’s tolerance levels
while designing the setpoint parameters for the DR algorithm and prioritize accordingly. For instance,
suppose that a blackout has occurred in certain regions of the grid. In order to reduce the stress imposed
on the grid, the EPU can set the aggregate demand setpoint to a very low value which can be sustained
over a short period of time during which ancillary services can be activated. On the other hand, if the
EPU desires to minimize dependence on expensive peak generation plants, then the aggregate demand
setpoint should be tuned to a higher value so that peak demand reduction can be sustained throughout
the day.
3.3 Demand Response via Population Games
In the previous section, a DR proposal based on the water-filling technique in convex optimization
has been presented. Although this technique exhibits desirable convergence characteristics, it is fairly
restrictive as load curtailment is considered to take continuous values. In the proposal presented next
(based on author’s work published in reference [95]), discrete power reduction levels are taken into
Chapter 3. Demand Response 39
account into the DR problem formulation, PEDR:
PEDR : minpr
fo(pr)
s.t.
m∑d=1
prd = SDR (C1)EDR
0 ≤ prd ≤ min {pd, bd} ∀ d = 1 . . .m (C2)EDR
prd ∈M = {M1 . . .Mn} ∀ d = 1 . . .m (C3)EDR
where m = |D| is the total number of consumers in the system. The objective function fo(.) is a
quadratic function (in accordance to Assumption 9) that applies a penalty for every power level in Mthat is curtailed. More specifically the structure of fo : Rn → R is:
fo(pr) =
1
2m
∑i∈M
( m∑d=1
Ii(prd))2
Aii
where Ii(prd) is an indicator function that returns 1 if prd = Mi and 0 otherwise and Aii is a strictly
positive value that is associated with the cost of demand curtailment Mi. This cost structure aims to
distribute the demand curtailment across the system so that excessive load reduction is not imposed
on one entity. PEDR is similar to the PDR formulation presented in the earlier chapter. In order to
overcome challenges introduced by discrete optimization variables in PEDR, theoretical constructs from
population games and convex optimization are utilized in the DR algorithm proposed next. The system
model and assumptions presented in Sec. 3.1 apply for this proposal as well. This is another distributed
solution in which the EPU transmits a general signal at every signalling iteration regarding the aggregate
state of the system which is then used by DR agents to make appropriate demand revisions based on
local operating conditions and available consumer budget. However, the method by which the EPU
signals and DR agent actuation are computed is distinct from the previous proposal. Fundamental
challenges present in PEDR due to intractability and highly varying local conditions are transformed
into advantages in this proposal. Discrete strategy space and a system consisting of a large number of
participants are fundamental building blocks in population game theory and therefore naturally applies
to PEDR.
3.3.1 Set of Transformations to the DR Problem
In this proposal, a set of transformations are applied to the DR formulation listed in PEDR. First,
all the local constraints (C2)EDR and (C3)EDR are removed from PEDR. These are then moved into
the corresponding DR agent’s local decision-making process which is discussed later. This results in
the first transformation P ′EDR. With P ′EDR, the EPU no longer needs to be concerned about local
demand constraints of individual participants. This significantly reduces communication overhead which
is otherwise required to maintain up-to-date information on individual demand parameters pd and bd.
P ′EDR : minpr
fo(pr)
s.t.
m∑d=1
prd = SDR (C1)EDR
Chapter 3. Demand Response 40
However, the EPU still requires information about individual power reduction prd of each participating
consumer in the transformed formulation. When there are thousands of participants in the system,
managing and keeping track of prd will require significant storage and computational overhead. In order
to address these concerns, another transformation is applied to P ′EDR with a change of variables by
noting that the main concern of the EPU is only with regards to the aggregate behaviour/costs in the
system and not individual decisions. This change of variables results in P ′′EDR:
P ′′EDR : minx∈4
fo(x)
s.t. m∑i∈M
Mixi = SDR (C1)EDR
where x belongs to a simplex:
x ∈ 4 = {x ∈ Rn |∑i∈M
xi = 1, xi ≥ 0}
The variables pr are replaced by x. The ith component of x represents the proportion of DR agents in
the system selecting power reduction level Mi. x can be computed from pr as follows:
xi =1
m
m∑d=1
Ii(prd)
When m → ∞, x can be considered to be a continuous variable as the discretization error 1m → 0.
Hence, with a large number of participants, x can be treated as a continuous variable which eliminates
the non-convexity issue encountered in P ′EDR due to the discrete nature of pr. Moreover, since n << m,
the optimization variable space of the transformed problem is much smaller. Next, the transformed
objective resulting from the change of variables is:
fo(x) =m
2
∑i∈M
x2iAii
which is a strictly convex cost function when Aii > 0. Strictly convex objective combined with the
linear coupling constraint and simplex conditions render P ′′EDR a strictly convex optimization problem.
Thus, solving P ′′EDR directly to obtain x∗ is straightforward for the EPU. The set of strategies available
to every DR agent is M. x∗ represents the optimal strategy distribution in the system to achieve the
lowest DR cost for the EPU. However, the main challenge arises when deciding how DR agents must be
allocated strategies in M to achieve this optimal x∗. As the EPU is not aware of the local operating
conditions and preferences of the consumers, this cannot be done centrally. It is necessary for the EPU
to transmit general information about the current state of the system repeatedly so that the DR agents
can iteratively revise their local strategies in a manner that achieves this optimal x∗ at equilibrium.
Chapter 3. Demand Response 41
3.3.2 Computation of EPU Signals
The computation of signals by the EPU that are broadcast to all DR agents at every signalling iteration
is presented next. First, the dual of P ′′EDR is constructed in P ′′EDRd:
P ′′EDRd : maxν
minx∈4
L(x, ν)
where L(x, ν) =m
2
∑i∈M
x2iAii + ν(SDR −m
∑i∈M
Mixi)
where ν ∈ R is a single-dimensional Lagrangian variable associated with the global coupling constraint
(C1)EDR. As the Slater’s condition is satisfied due to Assumption 8, the optimal solution of the dual
will be the same as the primal problem. x∗ can be computed directly by the EPU from the primal
formulation in P ′′EDR. This can then be used to solve for ν∗ in P ′′EDRd. The gradient of L(x, ν∗) when
x is not the optimal strategy distribution in the system is:
Fi(x) =∂L(x, ν∗)
∂xi= m(xiAii − ν∗Mi) i ∈M (3.2)
The signal transmitted by the EPU at every signalling iteration is selected to be F (x) = [Fi(x) . . . Fn(x)]T
which is computed using the current strategy distribution x in the system. This aggregate measure x
can be obtained from data concentrators (i.e. Assumption 1). The gradient F (x) can be interpreted
as the potential of the system. The DR agents that perform local strategy revisions use these signals
to react in a manner that reduces the potential of the system and thereby moving the system towards
the optimal strategy distribution. Moreover, it is important to note that the simplex condition on x is
automatically satisfied by the distributed solution model as all participating DR agents make a strategy
selection from one of the n available strategies.
A diagram illustrating this is presented in Fig. 3.7. In this example, there are three strategies (i.e.
|M| = 3) and therefore the simplex4 representing the values taken by x is a plane resembling a triangle.
Level sets associated with L(x, ν∗) are included in this diagram. The optimal solution x∗ lies at the
centre of the level sets and this represents the system state that will result in the lowest cost for the
EPU. Suppose that at a particular time instance, the system state is located at the point marked by
the black tangent line. The gradient at this point is denoted by the purple arrow. This is computed
by the EPU using Eq. 3.2. The DR agents must react so that the system state moves in a direction
that reduces the system potential (i.e. in the opposite direction of the gradient) towards the optimal
equilibrium as indicated by the orange arrow. With these strategy revisions that reduce the potential of
the system while accounting for the local operating conditions, the equilibrium distribution x∗ is in fact
the same as the optimal solution of the original formulation in PEDR as long as m→∞ and Assumption
8 holds. This property is demonstrated both theoretically and practically later in this chapter. It is
important to note that x∗ is unique, however, the optimal solution prd∗ that translates to this x∗ is not
unique. This is mainly due to the fact that the objective function fo(pr) is convex (not strictly convex).
Moreover, there are many non-unique combinations of prd that can be taken by individual consumers
that can result in x∗.
Chapter 3. Demand Response 42
Figure 3.7: Simplex and system potential.
3.3.3 A Game Theoretic Perspective
As per the transformations applied to PEDR detailed in the above, the state of the system is captured
by x, the strategy set available to DR agents is M and the cost assigned to each strategy i ∈ M is
Fi(x). These three elements completely define a game G(F, x,M). The players in the game constitute
of all DR agents participating in the DR program. Due to Assumption 7, as a large number of players
equipped with discrete strategy set are participating in the system, G defines a population game [96].
As the EPU exercises control over the cost Fi(x) of each strategy Mi, it dictates the behaviour of DR
agents in the system indirectly through these cost signals. Due to the specific structure of F (x) defined
in Eq. 3.2, G is also a potential game as the following externality symmetry is satisfied by the strategy
costs [96]:∂Fi∂xj
=∂Fj∂xi
, ∀ i, j ∈M (3.3)
The above relation holds for the F (x) defined in Eq. 3.2 for G as:
∂Fi∂xj
= 0,∂Fj∂xi
= 0 ∀ i, j ∈M, i 6= j
and thereby satisfying the externality symmetry conditions in Eq. 3.3. Players adjust individual strate-
gies by factoring into their decision-making process the F (x) transmitted by the EPU. When the players
reach a point where any other strategy revisions incur higher costs, the system has reached Nash equi-
librium (NE) defined as [97]:
xNE = {x ∈ 4|xi > 0→ Fj(x) ≥ Fi(x); ∀ i, j ∈M}
It can be shown that xNE for G is unique and exactly the same as x∗ which is the optimal solution of
P ′′EDR as detailed next. x∗ is a unique and optimal solution as P ′′EDR is a strictly convex optimization
Chapter 3. Demand Response 43
problem. KKT conditions associated with P ′′EDRd
are:
L(x, µ, λ) = fd(x) + µ(1−n∑i=1
xi)−n∑i=1
λixi
(1) Stationarity:∂L
∂xi= 0; Fi = µ∗ + λ∗i ∀ i ∈M
(2) Primal Feasibility:
n∑i=1
x∗i = 1, x∗i ≥ 0 ∀ i ∈M
(3) Dual Feasibility: λ∗i ≥ 0 ∀ i ∈M
(4) Complementary Slackness: λ∗i x∗i = 0 ∀ i ∈M
where fd(x) = L(x, ν∗). It can be shown that any x∗ satisfying the above KKT conditions also satisfies
the necessary and sufficient conditions required for an NE. For all strategies in use (i.e. x∗i > 0), condition
4 requires that the corresponding λ∗i = 0. This results in condition 1 reducing to Fi = µ∗ where i ∈ Mare all strategies in use (i.e. x∗i > 0). This implies that the cost of all of these strategies in use are the
same and is µ∗. For all other strategies j not in use, from condition 1 it is evident that Fj = µ∗+λ∗j and
it can be deduced that µ∗ + λ∗i ≥ µ∗ as λ∗i ≥ 0 due to condition 3. As the cost of these strategies are
higher, these are not in use. This shows that the costs incurred by the incumbent strategies are indeed
minimal and the same at optimality. Hence, the unique x∗ satisfying the KKT conditions is also the NE
of the game G. This establishes an equivalence between the DR optimization formulation and the game
theoretic equilibrium.
3.3.4 Resilience of EPU Signals
As DR agents rely upon cyber signals to make local demand curtailments, irrational behaviour by a
subset ε of these agents should be implicitly sensed by other unaffected agents via the EPU signals and
these will then react in a manner that returns the system to the optimal equilibrium state. The system
state xESS exhibiting this robustness is referred to as the evolutionary stable state (ESS) in population
games. It is next shown that xESS is in fact the same as xNE and therefore equal to x∗. This implies
that the optimal equilibrium in the system is robust to perturbations. As the game G has a strictly
convex potential function, it satisfies the relation [81]:
(y − x)′(F (y)− F (x)) > 0 ∀ x 6= y
which is a condition necessary for a strictly stable game. For x∗ to be an ESS, it must satisfy the
following conditions [98]:
(y − x∗)′F (x∗) ≥ 0 and if (y − x∗)′F (x∗) = 0 then (y − x∗)′F (y) > 0
The first condition dictates that x∗ is an NE according to an alternative definition of NE in reference [97].
It has been demonstrated in the previous section that this is indeed the case. As G is a strictly stable
game, the second condition is also naturally satisfied when (y − x∗)′F (x∗) = 0.
Chapter 3. Demand Response 44
3.3.5 Strategy Revisions by DR Agents
So far, the manner in which the EPU computes the signals that it periodically broadcasts is discussed.
In addition, the equivalence between the optimization and game theoretic framework has been presented
along with interesting insights into the equilibrium properties of the system. All of these static properties
will hold only when DR agents execute strategy revisions that reduce the potential of the DR system.
The strategy revisions made by each DR agent is dictated by the revision protocol ρi,j(x) which is a
function of the most recently received strategy cost F (x). ρi,j(x) defines the probability at which a DR
agent will switch from strategy i to strategy j where i, j ∈M. Prior to making the strategy switch, the
DR agent will ensure that local operating conditions such as consumer comfort budget and appliance
operating ranges are heeded. If these conditions are not satisfied, then the DR agent will not make a
strategy revision. This process ensures that the feasibility set defined in the optimization problem PEDRis always maintained by the distributive revisions.
Table 3.3 summarizes the strategy revisions made by a DR agent according to the afore-mentioned
procedure. In this algorithm, at the beginning of a day, the DR agent d initializes Ed to Edbudget. li is
the sum of reducible power consumption by active appliances that have been given permission by the
consumer for demand curtailment. Each DR agent randomly selects a strategy from y at the beginning
of the first signalling. Then in subsequent signalling iterations, DR agent selects an exponentially
distributed random time τi to revise its current strategy. When the revision time arrives, the DR
agent updates Ed and pd accordingly and proceeds to revise its current strategy using the latest signal
broadcast by the EPU. This is repeated until the current strategy cannot be further revised without
incurring more cost and/or the depletion of comfort budget. The EPU stops broadcasting the cost
signals when the optimal x∗ is achieved by distributive strategy selections in the system. The EPU will
be able to infer this via Assumption 1.
Distributed Algorithm for DR Agent i
Initialization:
• Optimization interval: ti ← 1; Next interval start time: ts ← ti
• Current time: tcurr ← 0, Ei ← Ebudgeti , tcong ← ti
• Current strategy used: sc ← y(1)
Algorithm (during congestion):
1. Compute τi using exponential distribution with rate 1. Settnext ← tcurr + τi which is the next strategy revision time
2. While tcurr < tnext and tcurr < ts:
• Set tcurr to the current internal clock time
3. Update:
• Ei ← Ei − sc min{τi, tcong}• If tcong < τi: ts ← ts + ti, tcong ← ti, τi ← 0
• tcong ← tcong − τi• pi ← Ei/tcong
Use the latest F broadcast by the EPU to select strategy scaccording to conditional switch rate defined by Eq. 3.4
4. sc ← {max{y}|y ≤ min{sc, pi, li}}5. Go to Step 1
Table 3.3: Distributed load curtailment via population game theory.
Chapter 3. Demand Response 45
The specific revision protocol considered in this algorithm leverages on pairwise comparison (PC)
and defined as follows:
ρPCi,j (x) = k[Fi(x)− Fj(x)]+ (3.4)
where k is a constant that normalizes ρPCi,j . During a revision, the DR agent will decide whether or not
to make a switch based on Eq. 3.4. If the cost of the current strategy is greater than the alternative
strategy, then the switching probability from strategy i to strategy j is positive. Otherwise, this is 0.
This switching protocol induces a dynamic in the system state x which is defined as follows:
xi =
n∑j=1
xjρj,i(x)− xin∑j=1
ρi,j(x) (3.5)
where the first term reflects the rate at which agents are switching into strategy i and the second term
reflects the rate at which the agents are switching out of strategy i. As ρ is a probabilistic measure, it
can be expected that the dynamics x of the system will exhibit some stochasticity. However, as there
is a large number of agents in the system, the strong law of large numbers (SLLN) will take effect and
eliminate stochastic effects. For this reason, Eq. 3.5 is referred to as the mean dynamics. This process
is also referred to as evolutionary game theory (EGT) as the system state x evolves due to changes in
F (x) as dictated by the EPU and local strategy revisions. The specific system evolution resulting from
ρPC is referred to as the Smith dynamic [96]:
xi = k
n∑j=1
xj [Fj(x)− Fi(x)]+ − kxin∑j=1
[Fi(x)− Fj(x)]+ (3.6)
The equilibrium of this dynamic should coincide with x∗ and this is discussed in the next section.
3.3.6 Theoretical Convergence Properties
The characteristics of the dynamic induced by DR agent strategy revisions listed in Eq. 3.6 determine
whether or not the system converges to the optimal strategy distribution x∗ and remains at that equi-
librium point. This is illustrated in Fig. 3.8. In this particular example, there are three strategies
available to agents. With the distributed strategy revisions, the agents should reach a stable equilibrium
which is also the optimal strategy distribution x∗. State dynamics that exhibit limit cycle behaviour
or consist of multiple equilibrium points are not desirable. With limit cycles, the system will not settle
and will keep on oscillating. With multiple equilibria, the system can converge to a point that is not
x∗. In order to determine whether the Smith dynamic results in desirable convergence characteristics,
concepts associated with negative correlation and (NC) nash stationarity (NS) which are defined below
are evoked [99].
NC: VF (x) 6= 0 implies that VF (x)′F (x) < 0
NS: VF (x) = 0 if and only if x ∈ NE(F )
where VF (x) is the system dynamic (i.e. right hand side of Eq. 3.6). When the first condition is satisfied,
the system behaviour is such that the growth rate of the population state is negatively correlated with the
corresponding system cost. Due to the potential game setup of G, the NC condition is naturally satisfied
Chapter 3. Demand Response 46
Figure 3.8: Illustration of system dynamics.
for the dynamics induced by the proposed algorithm [96]. When the system exhibits no limit cycles and
converges to equilibrium in a guaranteed manner, the following Lyapunov condition is satisfied [100]:
d
dtL(x) ≤ 0,
∂L(x)
∂x
∂x
∂t≤ 0, F (x)x ≤ 0
where L(x) is a Lyapunov function. Setting L(x) to fd(x) (this is the objective of P ′′EDRd
in which
ν = ν∗) and applying this to the above Lyapunov condition results in:
d
dtL(x) =
∂L(x)
∂x
dx
dt= F (x)x = VF (x)′F (x)
The last term is exactly the NC condition. Hence, as the Lyapunov condition holds for this system, the
state dynamic converges to the equilibrium with no limit cycles. The equilibrium of the dynamic defined
in Eq. 3.6 is in fact xNE as VF (x)′F (x) = 0 only when x = xNE and thereby satisfying NS. This implies
that the equilibrium of the dynamic is also the optimal solution x∗. Moreover, since xESS is the same
as x∗, the equilibrium of the distributed revisions is robust to perturbations.
3.3.7 Numerical Results
In this section, the theoretical results presented in the discussion above are ascertained via numerical
studies. Appliance penetration and usage parameters defined in Sec. 3.1.3 are leveraged to simulate
the load patterns of 1000 residential homes (i.e. m = 1000). Demand curtailment in each home can be
selected from the set M = {0.00001 0.1 1} kW. The cost aii associated with load curtailment i ∈ M is
set to a random strictly positive value increasing with i. In order to capture various efficient modes that
are made available in appliances by manufacturers, during conservation/demand curtailment periods,
appliances with resistive elements are randomly configured to reduce power consumption up to 90%.
Power consumption by cyclic inductive appliances such as fridges and freezers are not adjusted as the
temperatures maintained by these are critical. During the summer, air conditioners configured to operate
Chapter 3. Demand Response 47
in a flexible manner are completed turned off when required until the local comfort budget is exhausted.
Convergence and Initial States
In the first set of results depicted in Fig. 3.9, the convergence properties of proposed algorithm applied
to a system of 1000 participants are assessed over one DR cycle (optimization interval of 1 minute).
All possible values that can be taken by the three dimensional state x are contained within the simplex
0
0.5
1
0
0.5
1
0
0.2
0.4
0.6
0.8
1
y1=0.00001 kW
Solution Trajectories for PC Revision Protocol
y2=0.1 kW
y3=
1 k
W
Figure 3.9: Pairwise comparison revisions.
depicted by the triangular plane in Fig. 3.9. The costs associated with these states are reflected by the
level sets where the higher costs are associated with warmer colors. The optimal state x∗ lies in the middle
of these level sets. State trajectories ensuing from distributed revisions and iterative signal broadcasts
that begin at randomly selected initial values always converge to the optimal strategy distribution x∗
and it is clear that there are no limit cycles. This reaffirms the theoretical convergence insights gained
from the NC and NS conditions.
Impact of Population Size
An important requirement for obtaining the optimal solution of PEDR via distributive revisions enabled
by the population game construction is that the number of participants in the system must be very
large (Assumption 7). Although this is naturally satisfied by the large number of residential consumers
present in the system, it is possible that restrictions stemming from the exhaustion of comfort budgets or
local appliance operation conditions may limit the total number of participants in the system. In these
cases, the SLLN will not hold and stochastic effects will come into play. In order to illustrate the effects
of this, Fig. 3.10a presents the impact on the aggregate demand curtailment ratio when the number of
participants is varied from 10 agents to 1000 agents. This ratio r =∑i∈D p
rd/S
DDR is varied according to
the number of agents |D| present in the system by adjusting the setpoint SDDR. It is clear that when the
number of agents in the system is low (i.e. 10 agents), there are noticeable oscillatory effects whereas
with larger number of participants, the system smoothly converges to equilibrium. This is expected as
a large system serves to cancel the stochastic effects due to SLLN. In general, the system performs well
Chapter 3. Demand Response 48
when the number of participating agents is large as this renders x to be a continuous variable. The exact
precision of x is dictated by the number of agents in the system. Moreover, every agent contributes to an
incremental change in the system potential. Fig. 3.10b illustrates the overall demand reduction achieved
with respect to Sbudget = 600kW when the proportion of DR agents available is varied in increments of
0.2. It is clear that greater the availability of agents is, the closer will the system be towards achieving
the EPU’s demand reduction goals.
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (Seconds)
Ratio
Impact of Number of Agents on Performance
1000 Agents
100 Agents
10 Agents
(a) Aggregate demand curtailment in a limited system.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Impact of Availability of DR Agents
Proportion of DR Agents Available
Pro
port
ion o
f A
imed R
eduction A
chie
ved
(b) Availability of DR agents.
Figure 3.10: Performance of EGT based DR scheme under system limitations.
Resilience to Perturbations
Next, the robustness of the proposed algorithm is evaluated in Fig. 3.11 over a single DR cycle. It is
evident from this figure that the system converges to an equilibrium within the first 5 seconds of the
optimization interval. Then, after 25 seconds into the DR cycle, 20% of the agents are forced to choose
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
Time (seconds)
x
Proportion of Strategies in Population
y=0kW
y=0.1kW
y=1kW
Figure 3.11: System subjected to perturbation.
Chapter 3. Demand Response 49
a higher load curtailment by a malicious cyber attacker. This act will exhaust comfort budgets faster
and entail in more costs for the EPU. In the results illustrated in Fig. 3.11, it is clear that even after
this disturbance, the system is able to recover soon as other DR agents are able to implicitly infer this
perturbation via the EPU signals and adapt local demand curtailment to offset this perturbation. The
system returns to the equilibrium established prior to the attack soon thereafter. This adaptive resilience
of the system reinforces the theoretical assertion that the optimal equilibrium is ESS and therefore robust
to perturbations such as this.
Aggregate Demands over a Day
Next, the aggregate demands of consumers over a day with and without the proposed DR algorithm in
place is presented next in Fig. 3.12. In Fig. 3.12a, the EPU attempts to maintain aggregate demands
0 2000 4000 6000 8000 10000 12000 140000
50
100
150
200
250
300
Time (Hours)
Aggre
gate
Dem
and (
kW
)
Aggregate Demand with and without DR
No DR
Setpoint
With DR
8 12 24
(a) Setpoint is 200 kW.
0 2000 4000 6000 8000 10000 12000 140000
50
100
150
200
250
300
350Aggregate Demand with and without DR
Time (hours)
Aggre
gate
Dem
and (
kW
)
No DR
Setpoint
With DR
8 12 24
(b) Setpoint is 100 kW.
Figure 3.12: Ability of the EGT based DR algorithm to maintain demands around a setpoint.
around 225 kW. It can be observed that with the proposed algorithm in place, the aggregate demands
are successfully maintained around this setpoint over the whole day. In Fig. 3.12b, the EPU attempts
to maintain aggregate demands at a much lower setpoint of 125 kW. It is evident that the system is
unable to sustain demands around this value throughout the day as observed in the previous proposal
involving the water-filling method. This can be mainly attributed to the rapid exhaustion of the comfort
budgets set by the consumer. Hence, it is important for the EPU to strike a balance between demand
curtailment goals and consumer tolerance levels.
Comparison with the State-of-the-Art
The proposed DR algorithm is compared with a sub-gradient (SG) method similar to that proposed in
reference [34] over one DR cycle. This SG method, based on dual decomposition, consists of parameters
such as α that must be tuned with respect to the current configuration of the system. For the particular
optimization interval considered in Fig. 3.13, α = 0.0024 results in extensive oscillations that can cause
damage to system components whereas α = 0.0001 results in very slow convergence. On the other hand,
PC revisions require no such tuning and exhibits rapid convergence to equilibrium.
Chapter 3. Demand Response 50
0 10 20 30 400
200
400
600
800
1000Comparison between PC and Subgradient Methods
Iterations
Ag
gre
ga
te P
ow
er
(kW
)
PC
Original Demand without DR
SG: α = 0.0001
SG: α = 0.0024
Figure 3.13: Comparison between PC protocol and SG method.
A comparison with proposals in the existing literature that leverage population game theory for DR
is presented next. Work in [32, 39] propose DR schemes that utilize population games for distributed
appliance re-scheduling over an extended period of time (i.e. not real-time). Reference [32] leverages
a replicator dynamic (RD) based revision protocol for appliance scheduling throughout the day. RD is
based on imitative revisions that can result in non-unique equilibrium points. Work in [39], on the other
hand, proposes day-ahead appliance scheduling via best-response dynamic. According to [96], there are
two main issues with this dynamic. First, its convergence characteristics cannot be established using
traditional analysis (similar to that presented in Section 3.3.6) as the trajectories consist of differential
inclusions. Secondly, as the state trajectories are not unique, the system state can cycle in and out
of NE. As robustness and stability are important considerations, best-response dynamic is not ideally
suited for real-time DR.
To draw more general conclusions about the proposed DR strategy with respect to the state-of-the-
art, comparisons are made against four performance metrics as summarized in Table 3.4. DR strategies
in the literature are divided into three general classes and these are centralized real-time, offline and
decentralized real-time. These classes differ from one another with respect to the time horizons over
which computations occur (i.e. hourly or day ahead intervals) and how participating DR entities are
coordinated (i.e. by a central entity or in a distributed manner).
Proposed Centralized Offline DecentralizedDistributed Real-time Real-time
Comm Cost O(1) O(n) O(1) O(n)(n DR Agents)
Forecast No Yes Yes NoError
Solution Yes No Yes YesOptimalityResilience Yes No Yes No
Table 3.4: Comparison of DR methods.
Chapter 3. Demand Response 51
First, communication costs of the DR schemes belonging to these classes are evaluated. Due to the
cyber-physical vision of the smart grid (Assumptions 1-5), EPU and DR agents are able to communicate
with one another to enable efficient DR. Communication overhead is directly proportional to the amount
of information exchanged. In Table 3.4, the O−notation (Big-Oh) is used to capture the limiting
communication costs in terms of n which represents the total number of DR agents that are actively
participating in load curtailment [101]. In the proposed DR technique, communication overhead is
constant (i.e. O(1)) and independent of the size of the DR problem as the EPU broadcasts general costs
F (x) which are the same for all DR agents at every signalling iteration. Hence, the EPU does not forge
individual downlink connections with every DR agent. Moreover, it uses the wireless channel which is
an inherently broadcast medium (Assumption 2) [61]. Thus, a single general broadcast of the cost to
all DR agents at every signalling iteration suffices. Additionally, no information exchanges are required
between the DR agents in the proposed DR technique.
Centralized real-time techniques such as reference [31] utilize daily updates from participating DR
agents about local states (e.g. temperature, appliance operation, etc.) for computations so that fluctu-
ations in demands can be captured accurately. Also, as the central coordinating entity transmits signals
containing information about how to adjust various local configurations of appliances (e.g. temperature
setpoint) for every DR participant, these are specifically tailored for each DR agent. Hence, as com-
munication occurs between the EPU and the DR agents in both the uplink and downlink directions,
at least 2n information exchanges are necessary. This results in the communication cost amounting to
O(n). Offline strategies, such as the day-ahead pricing schemes referenced in [19], use only forecast
models (i.e. no need for information from consumers) to compute the price of electricity. This price is
common to all participating consumers and therefore can be broadcast. Hence, the communication cost
is constant O(1) for these techniques. Finally, decentralized real-time schemes such as the consensus-
based strategy proposed in reference [33] require information exchanges between participating agents,
the communication complexity is listed to be O(n).
Error from forecast is the second performance metric considered in Table 3.4. This error is non-
existent in the proposed and decentralized real-time schemes as prediction models are not employed.
Centralized real-time schemes are typically conducted over hourly intervals and offline schemes are
based on day-ahead forecasts which are associated with prediction errors [102].
Solution optimality is the third performance metric taken under consideration. For the proposed
DR method, this is guaranteed as demonstrated earlier via theoretical proofs and simulations. This
is also the case with decentralized real-time solutions as detailed in reference [33]. Centralized real-
time solutions may not be optimal as overcoming computational overhead which increases exponentially
with the number of participants due to discrete demand curtailment possibilities can be difficult with
time limitations. Heuristics are typically employed to resolve such issues and these have no guarantee on
solution convergence or optimality. On the other hand, as offline schemes typically use facilities equipped
with powerful computational resources with no time deadlines, optimality will be feasible.
Resilience is inherent in the proposed DR strategy as demonstrated via theoretical ESS and simulation
studies. Decentralized systems are vulnerable to cyber attacks as false information can be propagated
by compromised agents to other agents. This can lead to incorrect demand curtailment decisions.
Centralized schemes that depend on bi-directional information transfer are subject to single point of
failure attacks (attack on the central coordinating entity) or false information attacks on measurement
data sent by consumers. In the proposed DR strategy, as the EPU is the only entity initiating all
Chapter 3. Demand Response 52
communications, stringent security mechanisms can be applied that can fortify the EPU from single
point of failure attacks. This is more economical than securing every single participating DR agent.
Offline schemes, on the other hand, depend on forecast models for the computation of pricing signals
and therefore are not dependent on feedback from participants. As pricing information can be made
available from redundant sources, issues stemming from single point of failure can be averted as well.
3.4 Remarks
In this chapter, two DR proposals have been presented. The first is based on water-filling in convex
optimization with a limiting assumption made with regards to the load curtailment values. The second
proposal takes into account discrete demand curtailment levels and utilizes population game theory for
DR. Both proposals are highly scalable and suited for real-time DR in the residential sector. These dis-
tributed solutions abstract local operating conditions from the central coordinating entity and thereby
redirecting simpler computations to local DR agents. Demand curtailment in both of these proposals
factor in consumer comfort requirements. Moreover, robustness and resilience of the system to perturba-
tions such as cyber attacks when the proposed DR algorithm is in place are also demonstrated. The DR
agents are able to adaptively react to various conditions in the system just based on the general signals
transmitted by the EPU. Thus, as demonstrated by the two proposals detailed in this chapter, consumer
flexibility can indeed be leveraged to achieve various system goals that include reducing dependence
on peak generation sources and/or coordinating loads to prevent stress on the grid during blackouts.
These proposals can be easily extended to include deferment of appliance operation to a later time. This
can be achieved by introducing an additional metric that accounts for consumer tolerance to appliance
operation postponement. In order to prevent payback effects from all appliances operating at the same
time, random polling can be introduced prior to initiating the operation of these deferred appliances.
This will be considered in future work.
Chapter 4
Distributed Generation Dispatch
Renewable DGs are sustainable energy sources that are attractive alternatives to traditional synchronous
generation systems as commonly deployed fossil-fuel based plants are associated with high carbon foot-
prints. In addition to being clean energy sources, DGs also aid with the reduction of transmission line
losses as these are typically deployed at close proximity to consumers. This results in an efficient transfer
of power. However, DGs are intermittent energy sources with highly fluctuating generation capacities.
As generation forecast models for these are associated with significant error margins, day-ahead economic
dispatch algorithms used to coordinate traditional synchronous generation plants cannot be directly ap-
plied to manage these. Thus, DGs typically operate at MPP tracking mode in which maximum possible
power available from generation is directly injected into the grid. However, the recent rise of DG pen-
etration in the power grid evident especially in DNs is contributing to line congestion which can lead
to the infringement of physical infrastructure limits. The consequences of this will be irreversible and
expensive damages and outages. Thus, treating DGs as non-dispatchable entities will no longer be viable
in the near future.
In order to overcome these issues, two distributed proposals for enabling real-time DG dispatch are
presented in this chapter. The first proposal solves the classical economical dispatch problem for large-
scale DGs in real-time. Dual decomposition and the sub-gradient method are evoked to decouple the
economic dispatch formulation into a distributed problem which is then solved iteratively. In the second
proposal, a large number of small-scale DGs residing in the DN are coordinated. In addition to balancing
aggregate demand with supply, voltage rise considerations are also taken into account to ensure that
physical network limits are heeded. For this, population game theoretic constructs are leveraged. This
proposal is more generally applicable and practical as the underlying physical grid constraints are taken
into account. Prior to introducing these proposals, the system model, the general distributed solution
structure and assumptions are presented next.
4.1 System Model
The system considered in both proposals include DNs, grid-connected microgrids, and solar and wind
farms as illustrated in Fig. 4.1. This is a cyber-physical system in which every generation source is
equipped with intelligence and the ability to communicate. Moreover, the coordinating entity which can
include the ISO or EPU will have access to real-time measurements made available by data concentrators.
53
Chapter 4. Distributed Generation Dispatch 54
Figure 4.1: System model for economic dispatch.
Both proposals presented in this chapter are distributed and a general overview of the solution structure
is discussed in the following.
4.1.1 Distributed Dispatch
Like the solution model presented in Chapter 3 for DR, in this chapter, economic dispatch is also
based on a distributed architecture. The local operating conditions of the DGs are abstracted from the
coordinating entity (say the EPU). The EPU will have access to the aggregate state of the system which
is used to design general signals that are then broadcast periodically at every signalling iteration to all
participating DGs in the system. These signals are then used by the DGs to make dispatch decisions
based on the feasibility of local operating conditions (e.g. the current generation capacity, voltage rise,
etc.). This decoupling of local conditions from the main dispatch formulation prevents concentrated
computational and communication overheads as the EPU solves a simplified problem based on aggregate
measures and the DGs will not need to transmit local generation conditions to the EPU. Furthermore,
as the optimization intervals are highly granular, significant flux in demand and supply will be captured
by the proposed dispatch techniques with high accuracy. This implies that there is no need to evoke
forecast models that are associated with high error margins. The convergence and scalability properties
of these signalling and actuation decisions depend on the algorithms designed to compute these.
4.1.2 Assumptions
All assumptions made in the system model are summarized here:
1. The EPU can monitor aggregate behaviour of the system via data concentrators;
2. The EPU is equipped with wireless transmitters that can broadcast signals to all participating
DGs;
3. Each DG is a dispatchable cyber-physical entity equipped with communication capability, access
to local state measurements and intelligence;
Chapter 4. Distributed Generation Dispatch 55
4. The demand and supply remains constant over the optimization interval;
5. Cost of generation by sustainable DGs is lower than that of purchasing power from the main grid;
6. Cost of dispatch is quadratic;
7. Steady-state properties of the system are considered;
8. This is a grid-connected system with access to negative spinning reserves, synchronous generation
and storage systems;
9. The system consists of a diesel generator (also a DG) sized for the system; and
10. Three phases in the DN are decoupled into identical single phase network.
The first three assumptions align with the cyber-physical vision of the smart grid. Assumption 4 facili-
tates real-time dispatch goals as the optimization interval is set to 10 minutes in the first proposal and
1 minute in the second proposal presented in this chapter. More accurate forecast models can be evoked
over these short intervals by individual generation systems [103]. Assumption 5 encourages the use of
DGs which are sustainable and renewable power sources in lieu of traditional synchronous generation
associated with high carbon footprint and costs. Assumption 6 is typical in dispatch problems [88].
Assumption 7 implies that transients in the grid are negligible. This is an acceptable assumption due to
the grid-connected nature of the system and the existence of spinning reserves in Assumption 8. This
allows for the computation of economic dispatch in steady-state. Assumption 8 ensures that there is
sufficient generation capacity, inertia and reactive power supply in the system to provide uninterrupted
power supply to consumers. This is practical due to distribution system automation already enabled
in the grid via protective relays and volt/var compensation devices. Assumption 9 allows for complete
independence of the system from traditional synchronous generation systems located in the main grid
for real power supply. The final assumption also made in existing literature like reference [74] allows for
the consideration of the radial DN as a balanced single phase network.
4.1.3 Dispatching DGs
Traditionally, DGs such as solar and wind energy sources are considered to be undispatchable as these
have highly intermittent generation capacities influenced by external factors (e.g., cloud cover, wind
speed, etc.). Computing economic dispatch for renewables in typical day-ahead markets by evoking
prediction models is not common practise as forecast errors introduced by these models, especially
across longer prediction horizons, are not negligible [103]. In this chapter, two distributed algorithms
solve economic dispatch over every optimization interval and this interval is also referred to as the
economic dispatch (ED) cycle. Constraints in the problem formulation consisting of generation capacity
limits are updated every ED cycle and this allows the EPU to capture intermittencies in generation at
a high granularity as dictated by the length of the ED cycle. Power dispatch by DGs at prescribed
levels (not MPP) between zero and the maximum available power is supported by recent advances in
inverter design [84] and other alternative mechanisms. For instance, the power dispatched by solar panels
can be controlled by adjusting the angles of photovoltaic arrays [85]. Novel methods to dispatch wind
generation are actively investigated in the existing literature and one such example is demonstrated
in reference [86] which leverages pitch control and static convertor control to enable dispatch of wind
Chapter 4. Distributed Generation Dispatch 56
generation. However, if a system operator desires to not waste power generated, then the DGs can be
deployed in tandem with storage systems which are sized for the DGs now commercially available for
both industrial and residential use in accordance to Assumption 8 [10]. Then, the power output levels
of these storage systems can be varied or ‘dispatched’ as necessary. Although the current technologies
that support the dispatch of DGs and storage systems are still at infancy, significant research efforts in
this area both in the academia and commercial markets will certainly expand the number of available
options in the near future [10].
4.1.4 System Model Implementation
Numerical studies consisting of realistic solar and wind generation models are used to ascertain the
practicality of the proposed dispatch algorithms. A sample system is illustrated in Fig. 4.2. These
Figure 4.2: Sample system model for DG dispatch.
DGs range from roof-top solar panels and micro wind turbines to large-scale energy sources available in
solar/wind farms. These reside at close proximity to consumers.
Solar Generation
Power generation capacities available to solar DGs are governed by external environmental factors that
affect solar irradiance such as cloud cover. In order to realistically simulate generation by these DGs,
hourly solar generation data available in reference [104] is first smoothed so that there are no abrupt
generation changes between every one hour interval. Then, random noise is added to the data to model
irregularities in solar irradiance due to differences in cloud cover in the region.
Wind Generation
Wind generation is affected by the wind speed and the specification of various physical attributes of the
wind turbine. Wind speed is commonly modelled using the Weibull probability density function [61].
The parameters of this function are shape and scale factors which are set to be 1.94 and 4.48 [12]. To
simulate wind power generation, the power curve defined as follows is utilized:
Pwind =1
2Aρθv3
Chapter 4. Distributed Generation Dispatch 57
where A is the cross-sectional area of the turbine rotors, ρ is the air density, θ is the efficiency of the
turbine and v is the wind speed computed using the Weibull probability density function. The specific
parameters used in the power curve are detailed in numerical studies conducted later in this chapter for
each one of the proposals presented.
Power Demands
Consumer demands are simulated using models and parameters presented in Sec. 3.1.3. Specifics
regarding the number of homes implemented in the simulations will be listed in the numerical studies
conducted for each proposal later in this chapter.
4.2 Distributed Dispatch via Dual Decomposition
The first proposal presented in this chapter is based on the author’s work published in reference [106].
This is a distributed economic dispatch solution based on dual decomposition and the sub-gradient
method which is a classic optimization technique (as presented in reference [107]) applied in the context
of coordinating DGs in real-time in our paper listed in reference [106]. This proposal does not take into
account voltage rise considerations as common in high-level dispatch algorithms in the literature [45,50,
51]. Moreover, the possibility of discrete generation dispatch levels is ignored. However, the presence of
storage devices coupled with DGs that can actuate continuous values of real power, will eliminate the
need for this relaxation. The second proposal presented in this chapter is more practically and generally
applicable as it takes into account the underlying physical grid constraints and the possibility of discrete
dispatch. The main challenge overcome by the current proposal introduced next is the management of
DGs with highly varying generation capacities over short optimization intervals.
4.2.1 Economic Dispatch Formulation
The economic dispatch problem formulation used in the design of the proposed algorithm is listed in
PpDD.
PpDD : minp
C(p) =∑g∈G
2Cgp2g
s.t.∑g∈G
pg −∑d∈D
pd = 0 (C1)dd
0 ≤ pg ≤ cg ∀ g ∈ G (C2)dd
where p = [p1 . . . p|G|] is a vector consisting of power generation values of all generation systems, Cg > 0
is the cost of generation by DG g, (C1)dd is the global coupling constraint balancing aggregate power
demand with generation supply, (C2)dd ensures that the dispatch pg by each generator remains within the
available generation capacity. pd and cg parameters are updated at the beginning of every ED cycle. As
DGs are primarily considered, ramping constraints and specific lower bounds on generation capacity are
not included. Moreover, discrete generation dispatch is also not considered in this problem formulation.
Hence, as the objective and constraints are convex, this is a convex optimization problem. Although
this problem can be solved directly analytically or via commercially available solvers, the main issue
with a central approach is due to the communication overhead incurred when transferring fluctuating
Chapter 4. Distributed Generation Dispatch 58
generation capacity information by individual DGs to the EPU at a regular basis. This will result in
extensive communication overhead and resource requirements.
4.2.2 Master and Agent Problems
In order to overcome these issues, a two-step transformation process is applied to PpDD. First, the dual
of PpDD is constructed:
PdDD : maxλi1≥0,λi
2≥0,ν;minpL(λ1, λ2, ν, p)
where L(λ1, λ2, ν, p) =∑g∈G
2Cgp2g −
∑g∈G
λg1pg +∑g∈G
λg2(pg − cg) + ν(∑g∈G
pg −∑d∈D
pd)
where λ1, λ2 and ν are Lagrangian multipliers (i.e. dual variables) associated with the upper and lower
bounds of (C2)dd and the global coupling constraint (C1)dd respectively. The optimal solution of PdDDwill be same as that of PpDD when strong duality holds. This is indeed the case due to Assumption 9.
As PdDD is separable, this is decomposed into the master problem which is solved by the EPU and sub-
problems that are each solved by individual agents residing in DGs. First, local constraints are moved
from PdDD into the sub-problem PgDD that is solved by DG agent g. This eliminates the Lagrangian
multipliers λ1 and λ2 associated with the moved local constraints. Then, all terms in the objective of
PdDD that correspond to generator g are grouped together and these terms form the objective of PgDD.
The resulting sub-problem for DG g is:
PgDD : gg(ν) = minpg
2Cgp2g + pgν
s.t. 0 ≤ pg ≤ cg
where ν is a common constant that is computed and broadcasted to all DG agents by the EPU. Solving
PgDD will be straight-forward for each DG agent as the variable space is one-dimensional (i.e. pg) and
this is a convex problem that can be directly solved by a quadratic solver. The slack variable ν is key to
ensuring that there is a balance between overall demand and supply in the system. To compute ν, the
EPU will need to solve the master problem PmDD which is obtained from PdDD:
PmDD : maxν
M(ν) =∑g∈G
gg(ν)− v∑d∈D
pd
gg(ν) is the optimal solution of PgDD for fixed ν. This problem, however, cannot be solved directly by
the EPU for the following reasons. First, gg(ν) is the optimal value of individual PgDD and these sub-
problems are dependent on local operating conditions. Although M(ν) is concave, it is not differentiable
due to the local constraints in each gg(ν). Moreover, as the EPU is abstracted from the local generation
constraints, it cannot directly compute gg(ν). In order to overcome these issues, the sub-gradient method
is used by the EPU to iteratively compute ν as follows:
νk+1 = νk + αk+1q(νk) (4.1)
where k is the current updating/signalling iteration, q(νk) is the sub-gradient of M(ν) and αk is the
step-size at iteration k. The sub-gradient q(νk) of a non-differentiable function is not unique but must
Chapter 4. Distributed Generation Dispatch 59
satisfy the following relation [107]:
M(vk+1) ≤M(vk) + q(vk)(vk+1 − vk) (4.2)
One possibility of such a sub-gradient is:
q(νk) =∑g∈G
p∗g(νk)−
∑d∈D
pd (4.3)
where the first term reflects an aggregation of the power dispatch decisions made by individual DGs
based on the νk last broadcasted by the EPU which is used to solve PgDD. This is indeed a sub-gradient
as Eq. 4.3 satisfies Eq. 4.2 as follows:
M(vk+1) = infp∈X
(∑g∈G
2Cgp2g + νk+1(
∑g∈G
pg −∑d∈D
pd))
≤∑g∈G
2Cgp∗2g (νk) + νk+1(
∑g∈G
p∗g(νk)−
∑d∈D
pd)
=∑g∈G
2Cgp∗2g (νk) + νk(
∑g∈G
p∗g(νk)−
∑d∈D
pd) + (νk+1 − νk)(∑g∈G
p∗g(νk)−
∑d∈D
pd)
= M(νk) + (νk+1 − νk)(∑d∈D
p∗d(νk)−
∑d∈D
pd)
where X is the feasible set representing values that can be taken by p. The sub-gradient q(νk) can be
easily measured by the EPU by subtracting overall demands from current aggregate generation (which
are optimal dispatch decisions made by the DG agents based on νk that was last broadcast by the EPU)
at the kth iteration. Once equipped with the current value of the sub-gradient, to compute the next
update of νk which is νk+1, the EPU will need to define the step-size αk+1. Three step-size methods
are considered and these are defined as follows [107,108].
Constant step-size
With constant step-size, αk is fixed to the constant c > 0. Hence, αk will not change across the iterations.
Square-Summable-but-not-Summable step-size
In this specific method of assigning values to αk, the infinite summation of the square of αk should
be finite (i.e.∑∞k=1(αk)2 < ∞). However, this is not the case for the infinite summation of αk (i.e.∑∞
k=1 αk =∞). One possiblity of αk that satisfies this is αk = c/(d+ k) where c > 0, d ≥ 0.
Dynamic step-size
The main idea behind the dynamic step-size method is that νk+1 must be closer to the optimal ν∗ than
νk. Thus, substituting Equation 4.1 into ||v∗ − vk+1||22 < ||v∗ − vk||22 and rearranging results in:
αk = β(Mk −M(vk))/||q(vk)||22 (4.4)
Chapter 4. Distributed Generation Dispatch 60
where 0 < β < 2 and Mk is an estimate of the optimal value of M . The dynamic step-size algorithm
iteratively updates αk to reach the target level Mk. In the literature (i.e. reference [109]), the target is
set to Mk = maxi∈1...k
M(vi) + εk and εk is updated according to the following:
εk+1 =
ρ εk, if M(vk+1) ≥Mk.
max{γεk, ε}, otherwise.
where ε0 = ε, γ < 1, ρ > 1 are constants. The update algorithm at the kth iteration attempts to improve
the former value by εk. If the result is greater than the current iteration, the value of εk is increased by
a factor of ρ. Otherwise, εk is decreased by a factor of γ as long as εk ≥ ε.
An improvement to this computation is the path-based incremental target level algorithm detailed in
reference [110] that computes Mk by accounting for possible oscillations. An additional variable ωk is
maintained to detect oscillations. This variable is updated as follows:
ωk+1 = ωk + αkmQ
where Q is an upper bound on |q(vk)| and m is a scaling constant. Eq. 4.4 is altered where q(vk) is
replaced with mQ. The target level value is set to Mk = Mreck + εk where Mrec
k and εk are updated as
follows:
if M(vk) ≥Mreck +
εk2
;
εk = εk−1
Mreck = M(vk)
ωk = ωk−1
otherwise if ωk > b;
εk = εk−1
2 ,
Mreck = max
i∈1...kM(vi)
ωk = 0
else ;
εk = εk−1
Mreck = Mrec
k−1
ωk = ωk−1
where b is a path length bound on ωk, and Mk and εk are updated when the function value reaches the
target level. At this point, Mreck is set to the latest value of the function or is updated when the path
length exceeds the upper bound b. This implies that εk is too high and therefore reduced by half. When
these conditions are not met, all variables remain unchanged.
4.2.3 Termination Criterion
In an ED cycle representing one optimization interval, iterative updates using one of the step-size update
methods presented in the above and sub-gradient measurements obtained from data concentrators are
continued until the pre-defined termination criterion is met. This termination criterion for PpDD is set
to q(vk) = 0. This implies that there is a balance between aggregate supply and demand in the system
at which point optimality is attained and the algorithm can terminate.
Chapter 4. Distributed Generation Dispatch 61
4.2.4 Summary of Proposed Dispatch Algorithm
Table 4.1 summarizes the proposed distributed dispatch algorithm based on dual decomposition executed
during a single ED cycle (one optimization interval).
Distributed Dispatch for a single ED cycle via DualDecomposition
Initialization: EPU set νk = 0 for k = 1
1. EPU broadcasts νk to all participating DGs
2. DG g computes dispatch strategy pg(νk) by solving Pgdd
3. EPU implicitly infers sub-gradient q(νk) by measuringthe surplus or deficit power supplied by the utility tothe system
4. EPU updates αk+1 and νk+1 according to a step-sizeupdate method and the measured sub-gradient
5. If termination criterion is met (i.e. q(vk) = 0), thealgorithm ends. Else, set k ← k + 1 and go to Step 1
Table 4.1: Distributed dispatch algorithm via dual decomposition.
4.2.5 Convergence Analysis
The iterative updates that are broadcast by the EPU must converge to the optimal dispatch solution.
This convergence is mainly dictated by the step-size update method evoked to compute νk at every
signalling iteration. It is important to note that the smallest difference dk between the optimal value
M∗ of the cost function in PMdd and the current value of M(vk) evaluated up to iteration k is bounded
by:
dk = M∗ − maxi∈1...k
(M(vi) ≤
R2 +∑ki=1(αi)2Q2
2∑ki=1 α
i
)where ||v1 − v∗||22 ≤ R and ||q(vi)||22 ≤ Q. If R < ∞ and Q < ∞, then dk is an upper bound on the
difference between the current and optimal value of the dual cost function. This is mainly due to the
following set of derivations [111]:
||vk+1 − v∗||22 = ||vk + αkq(vk)− v∗||22 =
||vk − v∗||22 + 2αkq(vk)(vk − v∗) + (αk)2||q(vk))||22≤ ||vk − v∗||22 − 2αk(M∗ −M(vk)) + (αk)2||q(vk)||22
where the condition M∗ −M(vk) ≤ q(vk)(v∗ − vk) which is true for concave functions is applied to
obtain the final inequality in the above. Applying this relation recursively and using the facts that
||v∗ − vk+1||22 ≥ 0 and∑ki=1 α
i(M∗ −M(vi)) ≥ (M∗ − maxi∈1...k
M(vi))∑ki=1 α
i will result in dk.
For constant step-size updates, dk reduces to Q2h/2 as k → ∞. Hence, the optimal solution lies
within this threshold of the computed solution. Square-summable-but-not-summable step-size rule
converges to the optimal solution. Since∑∞i=1 α
2i < ∞ and
∑∞i=1 αi = ∞, dk → 0, this step-size
method converges to the optimal solution as k → ∞. A necessary condition for convergence of the
dynamic step-size computation is M∗ < ∞. The sub-gradient considered in this work is bounded as
Chapter 4. Distributed Generation Dispatch 62
Q = max{∑d∈D pd,
∑g∈G cg−
∑d∈D pd}. Since Q is bounded, the sub-gradient is also bounded. There-
fore, a necessary condition for dynamic step-size update convergence is met [112].
4.2.6 Numerical Results
In order to verify the theoretical conclusions presented earlier, the proposed algorithm is implemented
via MATLAB in simulations conducted using models presented in Sec. 4.1.4. The power rating of
the solar DGs is based on data supplemented for 3.08 kW panels [104]. The wind turbines considered
in these studies are assumed to have 50 kW power rating, 25m height, 13 m rotor diameter and 0.4
efficiency [105]. The generation sources supplement power for 200 homes. The diesel generator is sized
at 4 MW and is treated as a DG. The cost of wind, PV and diesel power generation is 0.135$/kWh, 0.802
$/kWh and 2.08$/kWh, respectively [43]. The cost of power supplied by the main grid is assumed to
be the same as that of diesel generation. Each ED cycle is 10 minutes in length. There are 30 signalling
iterations during this 10 minute period.
Comparison of Step-sizes
In the first set of results, the DG mix included in the simulations are 2 wind turbines, 20 solar panels and
one active diesel generator. The convergence behaviour of the system for the three step-size methods in-
troduced earlier is presented in Fig. 4.3 over three consecutive ED cycles. Various parameters associated
with these step-sizes are altered and the results are plotted for constant, square summable not summable
and dynamic step-size methods in Figs. 4.3a, 4.3b and 4.3c respectively. The solid blue line represents
aggregate demand in the system, all other curves represent aggregate real power dispatched by DGs
based on the proposed dual decomposition method for various combinations of step-size parameters.
From these results, it is evident that these methods converge to optimality. However, fine-tuning of var-
0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
12x 10
4 Constant Step Size
Time (hours)
Po
we
r (k
W)
c=1
c=2
c=10
Actual Demand
(a) Constant step-size.
0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
12x 10
4Square Summable but not Summable Step Size
Time (hours)
Po
we
r (k
W)
c=10, d=0
c=10, d=2
c=10, d=10
Actual Demand
(b) Square summable not summable.
0 0.1 0.2 0.3 0.4 0.50
5
10
15x 10
4 Dynamic Step Size
Time (hours)
Po
we
r (k
W)
b=6
b=7
b=8
Actual Demand
(c) Dynamic step-size.
Figure 4.3: Power dispatched by DGs for the three types of α computations.
ious parameters associated with these step-size computations is necessary to achieve rapid convergence
to optimality.
Aggregate Dispatch
Next, the ability of the proposed DG dispatch algorithm to distributively meet aggregate demands over
a day while accounting for generation fluctuations is investigated in Fig. 4.4. The dynamic step-size
Chapter 4. Distributed Generation Dispatch 63
update method is used in the updates of ν for this set of results. In Fig. 4.4a, the aggregate generation
capacity of the DGs (not including diesel generation) is plotted over a 24 hour period. It is clear that
the generation capacities are significantly fluctuating over the day. In Fig. 4.4b, the aggregate dispatch
of DG generation coordinated by the proposed algorithm is plotted alongside with overall demand in
the system. It is evident that the distributed dispatch of DGs closely matches the aggregate demands
in the system as expected.
0 5 10 15 200
0.5
1
1.5
2
2.5x 10
5
Time (hours)
Pow
er
(kW
)
Dispatch by 2 Wind Turbines and 20 PVs
Power Generated
Actual Demand
(a) DG generation without diesel generator.
0 5 10 15 200
0.5
1
1.5
2
2.5x 10
5
Time (hours)
Pow
er
(kW
)
Dispatch by 2 Wind Turbines, 20 PVs
Power Generated
Actual Demand
(b) DG generation with diesel generator.
Figure 4.4: Real-time dispatch over a day.
Sudden spikes in DG dispatch present in Fig. 4.4b are caused by transitions between one ED cycle
to another. Changes in generation capacities and aggregate power demands in the system can cause
the proposed algorithm to over-compensate and thus lead to the surges observable in Fig. 4.4b. This
is expected especially for the dynamic step-size update method as this method is associated with fast
convergence accompanied by initial oscillatory behaviour as illustrated in Fig. 4.3c.
Impact of Generation Mix
In the final set of results presented in Fig. 4.5, the impact of varying DG generation mix on the overall
power saved (i.e. power demand supplemented by not depending on the main grid or diesel generator)
is investigated. In Fig. 4.5a, various generation mix of solar panels and wind turbines are presented as
tuples in the x-axis. The overall power supplementing potential of these generation mixes are plotted
in this figure. It is clear that as the total number of generation in the DG mix increases, the greater
is the aggregate generation capacity of the system over the day. Next, in Fig. 4.5b, the regression of
power saved over the day for various generation mix combinations is explored. Solar panels, also denoted
as PVs, contribute to the highest generation around noon and wind turbines contribute more over the
evening. It is clear that significant rise in generation is evident during mid-day. Expensive peak power
can be supplemented by these sustainable generation sources. This regression analysis will be useful to
policy-makers when designing the integration of renewable generation into the grid.
Chapter 4. Distributed Generation Dispatch 64
(2,20) (2,40) (6,20) (6,40) (10,20) (10,40)0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6x 10
14 Generation Mix Impact
Generation Mix
Overa
ll P
ow
er
Saved (
kW
)
(a) Generation Mix Impact on Power Savings.
0 5 10 15 20 250
0.5
1
1.5
2
2.5x 10
8 Impact of Generation Mix
Time (hours)
Regre
ssio
n o
f P
ow
er
Saved (
kW
)
2WT, 20PV
2WT, 40PV
6WT, 20PV
6WT 40 PV
10WT, 20PV
10WT, 40 PV
(b) Regression of power savings over a day.
Figure 4.5: Power Savings.
4.3 Distributed Dispatch via Population Games
So far, a distributed economic dispatch solution has been proposed for optimally balancing aggregate
load and generation across short optimization intervals. Next, another distributed proposal is presented
which, in addition to being highly scalable with rapid convergence properties, also accounts for discrete
generation and underlying non-convex physical grid constraints. It is important to take into account
physical grid limitations as existing DNs are designed to accommodate power flowing from the main
grid to the consumers and not the other way around [113]. Due to significant penetration of DGs
which are located at close proximity to consumers located within a DN, physical network limitations
must be accounted for while integrating these. Otherwise, expensive infrastructure upgrades will be
necessary to prevent equipment damages. The economic dispatch proposal presented next allows for
safe coordination of large number of DGs present in radial networks such as the DN which are sensitive
to voltage rises resulting from excessive DG generation. Thus, this proposal generalizes the scope of the
dispatch formulation by leveraging on theoretical constructs from convex optimization and population
game theory. Details presented next are based on the author’s work published in references [114,115].
4.3.1 Dispatch Formulation
The problem formulation PEDV considered in this proposal is similar to that presented in PV R from
Chapter 2. This particular proposal is designed for DNs that are grid-connected (Assumption 8) and
focuses on real power dispatch by DGs. Hence, reactive power demands and line losses in the DN are
Chapter 4. Distributed Generation Dispatch 65
supplemented for by the main grid.
PEDV : minp
C(p) =∑g∈G
fg(pg)
s.t.∑g∈G
pg −∑d∈D
pd = 0 (C1)EDV
0 ≤ pg ≤ cpg ∀ g ∈ G (C2)EDV
pg ∈M = {M1 . . .Mng} ∀ g ∈ Gd (C3)EDV
|Vmin|2 ≤ VaV ∗a ≤ |Vmax|2 ∀ a ∈ B\0 (C4)EDV
Va = |Vref |∠0 a = 0 (C5)EDV∑g∈Ga
pg −∑d∈Da
pd + i(∑g∈Ga
qg −∑d∈Da
qd) =∑
a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b) ∀ a ∈ B\0 (C6)EDV
pm −∑d∈Da
pd + i(qm −∑d∈Da
qd) =∑
a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b) a = 0 (C7)EDV
The slack bus is set to be bus 0 and this is used as a point of reference. The substation bus where the
DN connects to the main grid is typically considered to be the slack bus. Voltage at the slack bus is
held to be constant value |Vref | with bus angle that is set to 0 (i.e. (C5)EDV ). (C1)EDV is a global
coupling constraint that enforces balance between overall distributed generation and demands. (C2)EDVto (C7)EDV are individual constraints that are local to every participating DG and buses in the DN.
(C7)EDV consists of pm and qm which are real and reactive power supplied by the main grid to meet
real line losses and reactive power demands in the system. In this formulation, individual DGs are not
modelled separately. These are represented by the main bus in the DN to which these physically connect
to (i.e. (C6)EDV and (C7)EDV ). This is a similar approach taken by other proposals in the literature
like references [74,113].
The main idea behind this proposal is to abstract local operating conditions from the EPU. This
process is illustrated in Fig. 4.6. The EPU will transmit general signals representing the aggregate state
of the power grid and the agents representing the DGs (Assumption 3) will respond to these signals
based on local operating conditions. Hence, the formulation presented in PEDV is decomposed into
master and secondary tiers. The EPU computes signals at the master tier and the DG agents compute
dispatch based on the received signals and local feasibility checks in the secondary tier. It is possible for
every DG to be represented by more than one agent and the reason for this will be discussed later.
In Fig. 4.6, a data concentrator aggregates all of the current strategies Si in use by DG agents and
overall power demands wi by consumers. The aggregated values are then transmitted to the EPU which
then uses these to compute signals that are then broadcasted to the DG agents.
4.3.2 Master Tier
In order to compute the signals broadcast by the EPU, a master problem is constructed by first moving
local constraints (C2)EDV to (C7)EDV in PEDV to the individual dispatch decisions made by the DG
agents. The resulting problem Pm′EDV is still associated with individual power dispatch values that can
Chapter 4. Distributed Generation Dispatch 66
Figure 4.6: System model for economic dispatch via master and secondary tiers.
be discrete.
Pm′
EDV : minp
C(p) =∑
pg∈Mg,g∈Gfg(pg)
s.t.∑g∈G
pg −∑d∈D
pd = 0 (C1)EDV
The EPU is only concerned about the aggregate behaviour of the system. A change of variables x ∈4 = {x ∈ Rn|xi ≥ 0,
∑ni=1 xi = 1} is applied to Pm′EDV . There are n components in x which are
associated with n power dispatch levels available to DG agents. These levels or strategies are denoted
as Y = [y1 . . . yn]. The ith component in x represents the fraction or proportion of agents in the system
using strategy yi. It is possible that the generation capacity of a DG is much higher than the largest
strategy level in Y. To prevent wasting the available generation capacity, the number of agents in a DG
is set to be mg = bcg/max(Y)c. Hence, the total number of agents in the system will be m =∑g∈Gmg.
The cost of power dispatch is represented by the function C(p). Since the cost of dispatch is typically
quadratic and higher penalty is levied for greater dispatch, C(p) is set to:
C(p) =
n∑i=1
(∑g∈G
mg∑j=1
Ii(pgj )pgj
)2
Ci
where pgj ∈ Y is the power dispatch selected by the jth agent in DG g, Ii(pgj ) is an indicator function that
returns 1 if pgj = yi and 0 otherwise and Ci > 0 is the cost assigned to power level yi. This cost structure
aims to distribute the power generation across the system so that excessive generation does not occur
in one part of the DN as this can result in over-voltage conditions. With this change of variables, the
Chapter 4. Distributed Generation Dispatch 67
resulting optimization problem for the EPU is PmEDV .
PmEDV : minx∈4
f(x) =
n∑i=1
(myixi)2Ci
s.t. m
n∑i=1
xiyi −∑d∈D
pd = 0 (C1)EDV
The dual of PmEDV is PdEDV :
PdEDV : minx∈4
L(x, ν) =
n∑i=1
(myixi)2Ci + ν(m
n∑i=1
xiyi −∑d∈D
pd)
where ν is the Lagrangian variable associated with the global coupling constraint (C1)EDV . The optimal
solution of PmEDV is identical to that of PEDV when m→∞ and the Slater’s condition is satisfied (due
to Assumptions 8 and 9). The EPU can directly compute x∗ and ν∗ as Pm′EDV is a strictly convex
problem with optimization variables x that are continuous as m → ∞ and the optimization variable
space is much smaller n << m. Although the EPU can directly compute x∗, the main difficulty arises
in assigning strategies to DG agents in the system to achieve this system distribution while ensuring
that local feasibility constraints such as generation capacity and bus voltage limits are met. From these
insights, the EPU signals are computed as follows:
Fi(x) = Kmyi(2Cimyixi + ν∗)
where F (x) = [F1(x) . . . Fn(x)]T is the cost of each strategy in Y that are available to the DG agents.
F (x) is in fact the gradient of L(x, ν∗). ν∗ is the optimal solution of PdEDV for fixed x∗ and K is a
constant used for normalizing as presented later on. The system state x, strategy set Y and strategy
cost F (x) define a game G(x, y, F ). Moreover, since a large number of DGs are considered, this is a
population game. The EPU will compute F (x) at every signalling iteration and broadcast this to all the
DG agents. The response by the DG agents is discussed next.
4.3.3 Secondary Tier
Each DG agent selects a random time to revise its current strategy which is selected from the set Y.
Dispatch Revision Protocol
Strategy revision by a DG agent is dictated by the revision protocol ρi,j(x) which is the probability of
switching from strategy yi to yj . The revision protocol defines the probability of a DG agent switching
from one strategy to another. Projection revision protocol listed in reference [96] is considered in this
proposal which is defined as:
ρi,j(F (x), x) =[Fi − Fj ]+
nxi(4.5)
This is a projection of the gradient of L(x, ν∗) onto the simplex 4. General mean dynamics is obtained
by subtracting the rate at which agents leave strategy i from the rate at which agents enter strategy i
Chapter 4. Distributed Generation Dispatch 68
as follows:
xi =
n∑j=1
ρj,ixj − xin∑j=1
ρi,j (4.6)
Substituting the projection revision protocol defined in Eq. 4.5 into Eq. 4.6 results in:
xi =1
n
n∑j=1
Fj(x)− Fi(x) (4.7)
For rapid reduction in the potential of the system, the system dynamic of x must evolve in a direction
opposite to this gradient and this is indeed the case with ρi,j .
Local Feasibility Voltage Checks
Prior to making a strategy switch, the DG agent ensures that local feasibility conditions hold. The
method by which this feasibility check is made is discussed next and illustrated in Fig. 4.7. There are
Figure 4.7: Local feasibility checks with voltage rise considerations.
two main feasibility checks that are conducted by the DG agent. First ensures that the DG’s generation
capacity limit is heeded and as this is a local check (i.e. (C2)EDV ). The discrete generation dispatch
level constraints in (C3)EDV are already handled by the discrete homogeneous strategy set Y made
available to each DG agent. The second check is to ensure that the current change in dispatch will not
cause voltage rise in the DN that will result in the violation of bus voltage limits (i.e. (C4)EDV and
(C5)EDV ). The bus voltages are governed by the power balance equations in (C6)EDV and (C7)EDVwhich constitute a set of non-convex quadratic equality constraints.
In order to ascertain that the voltage rise (if any) is within acceptable bounds, two main insights are
leveraged. First is that every DG agent selects a random time to revise its strategy. As the probability
Chapter 4. Distributed Generation Dispatch 69
of another agent selecting exactly the same time for revision is very low, it is possible to conclude that
each agent revises its strategy one at a time (i.e. not simultaneously). The second insight is that the DN
is a radial network which has a tree structure. Increase in generation will, therefore, only affect the local
feeder branch in which the revising DG agent resides. Based on these observations, the local feasibility
check conducted by each revising agent is discussed next and summarized in Tables 4.2 and 4.3.
Strategy Selection Process by DG Agent i
Initialization:
• Initialize time: tnext ← 0, t← 0.
• Initiate strategy: sc ← rand(y).
Start Algorithm: (repeat the following):
1. Time for next revision τi is computed via exponentialprobability distribution with rate µ. Next revision willoccur at tnext ← t+ τi.
2. When t > tnext,
• Utilizing the latest F and x broadcast by theEPU, select strategy si according to ρi,j(x).
• Set: sc ← si only if pi − sc + si ≤ ci.– Check if sc is feasible according to Table 4.3.
– If not, set sc ← si−1 and repeat previousstep.
– Else, go to Step (1).
Table 4.2: Strategy revisions by DG agents with voltage rise considerations.
The revising agent will need to ascertain whether the change in power dispatched ∆p = yi − yj
accompanying the strategy switch from yi to yj will have any impact on bus voltages of the local feeder
branch in the DN. Let the bus that the revising agent is currently residing at be denoted as b and the
parent bus as a (i.e. a ↔ b ∈ L and bus a is one line closer to the substation). Agents in bus a can
approximate the power flow pnewa,b in line a↔ b due to this change in dispatch using current power flow
denoted by polda,b obtained from local measurements (Assumption 3) as follows:
pnewa,b ≈ polda,b + ∆p (4.8)
As power loss in the line is not accounted for in this update, this is an approximation that allows
for a conservative estimation of voltage rise and thereby incorporates margins that can accommodate
unexpected transients. This impending change in power ∆p is communicated up the feeder branch until
all ancestors of bus b (up to the EPU/substation) update local power flows according to Eq. 4.8. This
upward propagation is referred to as the backward sweep as illustrated in Fig. 4.7. In the worst case,
the total number of information exchanges that can take place in the backward sweep step is the height
of the feeder branch when the revising agent is physically located in the last level of the feeder.
Then, a forward sweep process as illustrated in Fig. 4.7 is initiated starting with the substation node
(labelled a) and its immediate child (labelled b) that is an ancestor of the bus at which the revising
DG agent is residing. The voltage of the substation node is typically held constant. Voltage at bus b is
updated using the approximated power flow computed previously in the backward sweep step and qa,b
obtained from local measurements. If pa,b > 0, a voltage drop will occur across line a ↔ b and Vb is
Chapter 4. Distributed Generation Dispatch 70
updated by solving:
Va(Va − Vb)∗Y ∗a,b = pa,b + jqa,b (4.9)
Otherwise, there is a voltage rise and Vb is updated by solving:
Vb(Vb − Va)∗Y ∗a,b = −(pa,b + jqa,b) (4.10)
As Va is fixed at the substation, Vb is the only unknown variable. After updating the voltage, bus b
checks whether the local voltage constraint met (i.e., constraints (C4)EDV and (C5)EDV in PEDV ).
When this fails, an alarm is broadcast through this feeder to alert the revising agent that the dispatch
change is not feasible. Otherwise, the descendants of bus b repeat this until either an alarm is evoked
or the leafs of the updating feeder branch are reached. In the first case, the revising DG agent will infer
that the feasibility check has failed and in the second case feasibility is confirmed. This algorithm is
summarized in Table 4.3.
Feasibility Check by DG Agent i of Dispatch Change ∆P
• Backward sweep: Let b ← i, where i is the bus in which DGagent i resides in. Repeat the following until parent node isthe substation:
– Set a← parent(b) and send ∆p to node a.
– Update pa,b according to Eq. 4.8. Set b← a.
• Forward sweep: Let a ← substation bus and b ← child(a) inthe updating feeder branch. Repeat until descendent bus is aleaf or alarm activation:
– If pa,b > 0, then solve for Vb using Eq. 4.9 otherwise useEq. 4.10.
– If (C4)EDV or (C5)EDV are violated, activate alarmacross the updating branch
– Otherwise, set a ← b and b ← child(a) and send Va tonode b.
Table 4.3: Voltage feasibility check by DG agents.
Next, the computational complexity of the forward and backward sweep method is analyzed. The
communication complexity of the feasibility check in the worst case is 3hn where h is the height of
the tree representing the DN, the constant 3 denotes three sets of information flow across the updating
branch (i.e. the forward, backward and alarm information exchanges) and n denotes the number of
strategies available for each DG agent. For instance, consider the low-voltage Danish DN consisting of
34 buses supplying power to 75 homes detailed in reference [116] amd depicted in Fig. 4.8 in which,
the height of the deepest feeder tree is 7 and the number of strategies available to agents is n = 3.
As the latency incurred by information exchanges is in the order of microseconds due to the physical
proximity of the nodes, the time required to check voltage constraints will be much less than 1 second.
Also, computational complexity at each agent is constant as the backward sweep involves one addition
and the forward sweep requires the solving of a quadratic equation in the worst case. In general, the
computations performed by each DG agent for the secondary tier involves determining ρi,j(x) from
the signals broadcast by the EPU, estimating local power flows and comparing dispatch strategies (to
Chapter 4. Distributed Generation Dispatch 71
ensure that generation capacity limits are heeded). All of these secondary tier computations entail simple
mathematical operations and thus are straightforward.
4.3.4 Equivalence between Game Characterization and Optimization
A population game G is defined by individual active players selecting a strategy from the set Y based
on the allocated cost F (x). x represents the proportion of players in the population selecting strategies
in Y. Suppose, there exists a function f : Rn → R such that:
F (x) = Of(x) ∀ x ∈ 4
then f is defined to be a full potential function for the game G. Moreover, if the following full symmetric
condition is met:∂Fi(x)
∂xj=∂Fj(x)
∂xi∀ x ∈ 4, yi ∈ Y, yj ∈ Y
G is a fully potential game in which revisions made by each player reduce the potential of the system.
Nash equilibrium x∗NE of this system is the same as the optimal solution of the following problem when
m→∞ [96]:
Pp : minf(x)
pgj ∈ y ∀ j = 1 . . .m
xi =1
m
m∑j=1
Iyi(pgj ) ∀ i = 1 . . . n
Due to Assumptions 8 and 9, reactive and real power losses in the lines are considered to be supple-
mented by the main grid where qm =∑d∈D qd−
∑a↔∈E q
la,b−
∑g∈G qg and pm =
∑a↔b∈E p
la,b thereby
simplifying PV R to PEDV reiterated in the following for convenience of reference:
PEDV : minp
C(p) =∑g∈G
fg(pg)
s.t.∑g∈G
pg −∑d∈D
pd = 0 (C1)EDV
0 ≤ pg ≤ cpg ∀ g ∈ G (C2)EDV
pg ∈M = {M1 . . .Mng} ∀ g ∈ Gd (C3)EDV
|Vmin|2 ≤ VaV ∗a ≤ |Vmax|2 ∀ a ∈ B\0 (C4)EDV
Va = |Vref |∠0 a = 0 (C5)EDV∑g∈Ga
pg −∑d∈Da
pd + i(∑g∈Ga
qg −∑d∈Da
qd) =∑
a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b) ∀ a ∈ B\0 (C6)EDV
pm −∑d∈Da
pd + i(qm −∑d∈Da
qd) =∑
a↔b∈L
(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b) a = 0 (C7)EDV
Chapter 4. Distributed Generation Dispatch 72
With the change of variables x and the notion of multiple agents per DG, PEDV can be equivalently
posed as:
P ′EDV : minx
n∑i=1
Ci(m.yi.xi)2 − ν∗(
∑d∈D
pd −mn∑i=1
yi.xi)
m =∑g∈Gbcg/max(Y)c (C1’)EDV
pjg ∈ {yi ∈ y|1Cj (yi) = 1} ∀ j = 1 . . .m (C2’)EDV
xi =1
m
m∑j=1
1yi(pgj ) ∀ i = 1 . . . n (C3’)EDV
where the coupling constraint (C1)EDV from PEDV is moved to the objective and this term will be 0 at
optimality as it is assumed that there exists sufficient generation capacity to meet overall demand. ν∗ is
a constant computed from PdEDV . The fully symmetric condition is satisfied by the objective of P ′EDV .
The indicator function ICj (yi) determines the feasibility of strategy yi based on constraints (C2)EDV
to (C7)EDV for DG agent j. Problems PEDV and P ′EDV are equivalent due to the change of variables
and feasibility checks in (C2’)EDV . m will depend on the availability of generation capacity during the
current optimization interval. When m→∞, x can be considered to be continuous. It is important to
note that C(p) in PEDV results in non-unique optimal solutions with the same optimal value. To see
this, consider∑ni=1 Ci(m.yi.xi)
2 which is equivalent to C(p) in terms of x. xi is computed by counting
the total number of agents in the population using strategy i divided by m. Suppose, x = {0.2, 0.3, 0.5}and m = 1000, then the total number of combinations of pig that result in this x is
(1000200
)(800300
). A subset
of these vast number of combinations result in the feasibility of ICj (yi), given that there exists sufficient
generation capacity in the system. Hence, there are many ways of arriving at x∗. Although x∗ is unique,
PEDV can lead to the same optimal x∗ with non-unique combinations of pig. Thus, with the projection
revision protocol and feasibility checks, DG agents arrive at a configuration not necessarily unique but
results in this x∗. Convergence characteristics and bounds to ensure that the feasibility checks are not
overly conservative are provided in the following.
4.3.5 Theoretical Convergence Properties
The distributed strategy selections by the DG agents should be such that the state dynamics converge
to the optimal solution of PEDV without any limit cycles. For this, the dynamic introduced by these
revisions listed in Eq. 4.7 must have a strict Lyapunov function. For rapid convergence to optimality, it
is necessary to demonstrate that the system converges to optimality exponentially fast. Moreover, the
approximations applied in the voltage checks must not be too conservative.
Exponential Convergence due to Projection Revisions
In order to show that the projection dynamic converges exponentially fast to the equilibrium x∗ (i.e.
||xt−x∗|| ≤ e−αt/2||x0−x∗|| for α > 0) where xt is the system state at time t and x0 is the initial system
state, it is necessary to show that there exists a Lyapunov function L(x) such that L(x) ≤ −αL(x) for
α > 0. The candidate Lyapunov function is selected to be L(x) = (f(x) − f(x∗)) + 12 ||x − x
∗||2 where
Chapter 4. Distributed Generation Dispatch 73
f(x) is the abbreviated form of the potential function L(x, v∗) and x∗ is the optimal solution of PmEDV .
Prior to discussing the proof, it is important to note that the cost F (x) = [F1(x) . . . Fn(x)]T is strongly
monotone as the following holds:
(F (x)− F (x∗))T (x− x∗) ≥ m(min(Ciyi)||x− x∗||2)
Let µ = m(min(Ciyi)) and it is evident that µ > 0 as Ci, yi and m are strictly positive. Moreover, as
the function f(x) is convex, the following holds as well:
(F (x)− F (x∗))T (x− x∗) ≥ f(x)− f(x∗)
Equipped with these properties, the following is a proof of exponential convergence of the projection
dynamic:
d
dt(L(x)) = OL(x)T x = (F (x) + (x− x∗))T x
= (F (x) + (x− x∗))T (1
n(
n∑j=1
Fj(x))1− F (x))
= −||F (x)− (1
n
n∑j=1
Fj(x))1||2 − (F (x)− F (x∗))T (x− x∗)
≤ −(F (x)− F (x∗))T (x− x∗)
where 1 is an n-dimensional vector of ones. The first term in the third line of the above proof is ≤ 0
and removing it has resulted in the last inequality. The fact that F (x∗) = 0 is also used to obtain the
third line. Combining the strong monotone condition of F (x) and the convexity of f(x), the following
inequality results:
(F (x)− F (x∗))T (x− x∗) ≥ 1
2(f(x)− f(x∗)) +
µ
2||x− x∗||2
≥ min(1
2, µ)((f(x)− f(x∗)) +
1
2||x− x∗||2)
= min(1
2, µ)L(x)
Hence substituting the above inequality into the last line of the initial proof, we are able to show that
L(x) ≤ −min(1
2, µ)L(x)
where α = min( 12 , µ) > 0 and thus the projection dynamic will converge to x∗ exponentially fast.
Bound on Voltage Rise
Here, a proof is presented on the upper bound of voltage increase due to increase in power dispatch by
∆P at bus b. Suppose that bus a is the direct ancestor of bus b (i.e. a ↔ b ∈ E). The relationship
between bus voltages and power flow across the lines is as follows:
Vb(Vb − Va) = (pb,a + iqb,a + ∆P )za,b
Chapter 4. Distributed Generation Dispatch 74
Vb and Va are the voltages resulting from the back flow of power in the line (i.e. pb,a + iqb,a + ∆P ) and
∆P is the increase in power dispatch across the line b↔ a. Since the above relation consists of complex
variables (i.e. Va, Vb, pb,a + iqb,a), the magnitude of voltage difference across the line a↔ b is:
|Vb − Va| =|(pb,a + iqb,a + ∆P )za,b|
|Vb|
By evoking the triangular inequality, the above is bounded by:
|Vb − Va| ≤|(pb,a + iqb,a)za,b|+ ∆P |za,b|
|Vb|
The first term on the right side of this relation is |(pb,a + iqb,a)za,b| = |V ′b ||V ′b − V ′a| where V ′b and V ′a
voltages at buses a and b prior to the change in dispatch. Substituting this, the following inequality is
obtained:
|Vb − Va| ≤|V ′b ||V ′b − V ′a|
|Vb|+
∆P |za,b||Vb|
As the dispatch increase will result in a voltage rise,|V ′b ||Vb| ≤ 1. Moreover, as the fixed substation voltage
is the point of reference and |Vb| ≥ |Vs|, it can be concluded that ∆P|Vb| ≤
∆P|Vs| . From these observations,
the above bound is further generalized to:
|Vb − Va| ≤ |V ′b − V ′a|+∆P |za,b||Vs|
Hence, the first term in the above inequality represents the voltage difference across the bus prior to
the increase in dispatch. The second term represents an upper bound on voltage rise contributed by the
increase in dispatch. As ∆P is small, the margin of error is also small.
4.3.6 Resilience
The system dynamic has been shown theoretically to have strong convergence characteristics due to
exponential stability. This property is also important in identifying how the system will react to per-
turbations introduces by cyber attacks or other disturbances. Due to exponential stability, even with
perturbations, the system state trajectory will always be attracted to the stable equilibrium point which
is also the optimal state. Hence, as long as there exists sufficient capacity in the system (i.e. generation
capacity and bus voltage limits), the DG agents that are not compromised will be able to sense these
changes implicitly through the EPU signals and react to offset the disturbances.
4.3.7 Numerical Results
Simulation studies are conducted using MATLAB to verify whether the theoretical results presented
earlier also apply in practical settings. The DN implemented is a low-voltage radial network illustrated
in Fig. 4.8 consisting of 34 buses and 75 homes.
Line impedance parameters are obtained from reference [116]. Bus voltage and reactive power flow
measurements used by individual agents are computed using Gauss-Seidel load flow analysis method
[117]. As the main intent of this proposal is to coordinate a large number of DGs, every home is
Chapter 4. Distributed Generation Dispatch 75
Figure 4.8: Low-voltage distribution network.
considered to contain a roof-top solar panel and a micro wind turbine. Power demands of homes in the
DN and generation by the DGs are modelled as outlined in Sec. 4.1.4. The specific parameters used
to model wind turbines are obtained for a power rating of 1.9kW as listed in reference [118]. The set
of strategies available to the DG agents is Y = [0.0000001 0.01 0.02] kW in these simulations. Costs
of these strategies are set to strictly positive random values which are increasing with respect to the
power dispatch levels (i.e. Ci < Ci+1). The voltage magnitudes are bounded by ±10% of the nominal
voltage of 1 p.u. (per unit). The forward and backward sweep steps require communication between
Figure 4.9: Topology of cyber-physical DN.
nodes along the local feeder branch. Hence, in this proposal, in addition to EPU broadcasting signals
Chapter 4. Distributed Generation Dispatch 76
at regular intervals, DG agents exchange information with a subset of other nodes in the system. The
overlay of the communication network with the physical DN is illustrated in Fig. 4.9.
Convergence over One ED Cycle
The convergence characteristics of the proposed dispatch algorithm is first assessed over one optimization
interval in Fig. 4.10. Various initial states of x at the beginning of an optimization interval are randomly
00.5
1
0
0.5
1
0
0.2
0.4
0.6
0.8
1
y1=0.000001 kW
State Trajectories for Single Dispatch Cycle
y2=0.01 kW
y3=
0.0
2 k
W
1500
2000
2500
3000
3500
4000
4500
Level Set
Optimal Solution of P
M
A State Trajectory
Figure 4.10: Projection revisions.
generated and the state trajectories induced by distributed revisions are captured in Fig. 4.10. The level
sets of L(x, ν∗) are included in this figure. The lowest possible value of the potential function L(.) is
located at the centre of these level sets as indicated by the teal circle. It is clear that the state trajectories
rapidly converge to the optimal state regardless of the initial state. Moreover, the state trajectories are
more or less orthogonal to the cost level sets. This indicates that the convergence is exponentially fast.
Next the impact of DG agents’ availability on the convergence properties of the system is examined
in Fig. 4.11. An important assumption made in this proposal is the presence of a large number of
DG agents which ascertains optimality and validates the use of population game theoretic constructs.
The plot located at the top left-hand corner of Fig. 4.11 shows that when only 20% of the agents
are available, there are significant oscillations in the demand and supply ratio which are introduced by
active stochastic effects. This is not the case for larger proportions of availability such as 50% and 100%.
Fluctuations in overall costs are not significantly evident due to the considerably large cost coefficients
that are in effect. The minimum and maximum voltages across the 34 buses in the system are plotted
in the bottom figure. It is clear that the bus voltages are maintained well within ±10% of the nominal
voltages limits.
Resilience to Perturbations
The resilience and recovery speed of the system in the event of perturbations to system state due to
cyber attacks is investigated in Fig. 4.12. It is possible for malicious entities to perpetrate attacks
Chapter 4. Distributed Generation Dispatch 77
0 0.005 0.01 0.0150
0.5
1
x 10−3Ratio of Real Power
Time (hours)
Dem
and/S
upply
Ratio
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0161.026
1.028
1.03
1.032
1.034
Min and Max Voltage in the DN during a Day
Time (hours)
Norm
aliz
ed V
oltage
Max Volt 20%
Min Volt 20%
Max Volt 50%
Min Volt 50%
Max Volt 100%
Min Volt 100%
0 0.005 0.01 0.0150
0.5
1
Cost
Time (hours)
Cost R
atio
Availability: 20%
Availability: 50%
Availability: 100%
Figure 4.11: Impact of population size.
0 0.005 0.01 0.0150
10
20Impact of Disturbance
Real P
ow
er
(p)
0 0.005 0.01 0.0150.9
1
1.1
Time (hours)
Voltage (
p.u
.)
Dispatch
Demand
Max VoltageMin Voltage
Figure 4.12: Recovery of DG Agents from Disturbance.
Chapter 4. Distributed Generation Dispatch 78
such as forcing DG agents to choose more expensive strategies (higher dispatch level) that can cause
unnecessary expenses and voltage rise issues. In Fig. 4.12, 20% of agents (i.e. 200 DG agents) have
been forced to select the highest power dispatch level possible 30 seconds into a dispatch cycle. At this
point, a spike in aggregate real power dispatch and maximum bus voltage is evident. However, other
DG agents immediately respond to this disturbance and adjust their local dispatch levels to bring the
system back to the optimal steady state operation. Since this is a distributed dispatch strategy with
agents making independent decisions, it is difficult for an adversary to compromise a large number of
agents.
Dispatch over a Day
The performance of the proposed algorithm over a period of 24 hours is evaluated next. In Fig. 4.13,
0 5 10 15 200
20
40
60
80
100
120
140
160Aggregate DG Capacity
Time (hours)
Real P
ow
er
(kW
)
(a) DG Capacity.
0 5 10 15 20
0.9
1
1.1
1.2
1.3
1.4Voltage Profile over a Day with no Dispatch
Time (hours)
No
rma
lize
d V
olta
ge
Max VoltageMin Voltage
Voltage Limits
(b) Voltage Profile Over 34 Buses.
Figure 4.13: System behaviour (real power and bus voltages) without DG dispatch in place.
no dispatch coordination mechanisms are in place and all power generated is directly injected into the
grid. The aggregate real power generation is plotted in Fig. 4.13a and the minimum and maximum
bus voltages are plotted in Fig. 4.13b over a period of one day. It is clear that the bus voltages are
exceeding the maximum voltage threshold at many points throughout the day. These over-voltages will
result in adverse equipment effects. Fig. 4.14 illustrates the behaviour of the system after applying
the proposed dispatch technique in the system. It is clear that the aggregate demands are closely met
by the aggregate generation in the system as depicted in Fig. 4.14a. Furthermore, as illustrated in
Fig. 4.14b bus voltages are maintained well within the ±10% threshold. These results demonstrate the
effectiveness of the proposed dispatch algorithm in coordinating a large number of DGs located in a DN
while accounting for physical infrastructure limitations. No fine-tuning of any parameters is required
and the technique is highly scalable, adaptive and flexible.
Comparison with State-of-the-Art
Next, the proposed dispatch strategy is compared with similar existing literature. More specifically, in
Fig. 4.15, a comparison is presented in which the proposed algorithm is compared with a sub-gradient
Chapter 4. Distributed Generation Dispatch 79
0 5 10 15 200
2000
4000
6000
8000
10000
12000
14000
16000
18000Aggregate Real Power Demand and Dispatch
Time (hours)
Real P
ow
er
(W)
DG Dispatch
Consumer Demand
(a) Real Power.
0 5 10 15 20
0.9
0.95
1
1.05
1.1
Min and Max Voltage Profile over a Day with Dispatch
Time (hours)
No
rma
lize
d V
olta
ge
Max VoltageMin Voltage
Voltage Limits
(b) Voltage Profile Over 34 Buses.
Figure 4.14: Real-time dispatch (real power and bus voltages) over a day with dispatch solution in place.
method based on dual decomposition. For sub-gradient methods, it is necessary to fine-tune various
parameters. Specifically in the implementation presented in Fig. 4.15, α is the parameter being tuned.
For large values of α, significant oscillations are present in the dispatch of real power and the bus voltages
as illustrated in Fig. 4.15a and 4.15b. These oscillations are significant and render the system conducive
to equipment damages. Moreover, if the oscillations are too high, the bus voltage limits can be violated
as well. With lower values of α, the convergence speed is too slow. The proposed algorithm, on the
other hand, requires no such fine-tuning and exhibits extremely fast convergence to optimality with no
oscillations.
0 20 40 60 80 1000
2
4
6
8
10
12
14
Signalling Iterations
Re
al P
ow
er
Dis
pa
tch
ed
(kW
)
Comparison of Distributed Dispatch Methods
Our Proposal
SG with α: 0.025
SG with α: 0.0005
(a) Real Power Dispatch.
0 20 40 60 80 100
1.02
1.025
1.03
1.035
1.04
1.045
1.05
1.055
1.06Min/Max Voltage Over a Day for Various Algorithms
Signalling Iterations
Vo
lta
ge
(p
.u.)
Min Voltage for Our Proposal
Min Voltage for SG α: 0.025
Min Voltage for SG α: 0.0005
Max Voltage for Our Proposal
Max Voltage for SG α: 0.025
Max Voltage for SG α: 0.0005
(b) Voltage Profile Over 34 Buses.
Figure 4.15: Comparison of Dispatch Methods.
Next, general characteristics of the proposed and existing dispatch strategies are evaluated against
four performance metrics (communication costs, information overhead, error from forecast and solution
optimality) as summarized in Table 4.4. The O-notation is used for quantifying communication costs and
Chapter 4. Distributed Generation Dispatch 80
information overhead with respect to the size of the systemm (i.e. number of participants). The proposed
distributed dispatch strategy is compared with three general classes of dispatch strategies in the existing
literature and these are centralized real-time (i.e. online), offline and decentralized online solutions.
Centralized online solutions use a central coordinating entity to compute dispatch in real-time (over
short optimization intervals). Offline solutions compute dispatch using generation prediction/forecast
models well in advance. Decentralized online solutions use peer-to-peer communications to compute
dispatch in real-time.
Proposed Centralized Centralized/ DecentralizedDistributed Online Distributed Online
Offline
Communication Costs O(hn) O(m) 0 O(m2)
Information Overhead O(hn) O(m) 0 O(m2)
Forecast Errors No Yes Yes No
Solution Optimality Yes No Yes Yes
Table 4.4: Comparison of dispatch methods.
Communication cost refers to the number of connection links forged by nodes in a particular strategy
at every dispatch cycle. For the proposed strategy, the utility does not make point to point connections
with the DGs but broadcasts a single transmission. During feasibility checks, the number of information
exchanges in the worst case is 3hn. Hence, the communication complexity is O(hn) where hn << m.
For offline strategies, as the entire dispatch schedule will be communicated in advance, communication
cost is negligible. For centralized online strategies, all m DG agents will need to communicate individual
generation capacities to the EPU that will then communicate m dispatch values to the DGs resulting in
a communication cost of 2m [43]. For decentralized algorithms, according to [61], communication cost is
m(m− 1) where m represents the total number of relays in the system which is assumed to be at least
as large as the number of DGs coordinated.
Information overhead accounts for the total units of data exchanged between the nodes participating
in dispatch. This is proportional to the communication costs detailed in the above for the proposed
dispatch strategy. The information overhead for offline strategies is ignored since exchange occurs in
advance. For centralized online strategies, DGs transmit current generation capacity and the central
controller transmits the computed dispatch vector to all DGs. Given that there are m DGs in the
system, this results in 2m units of information exchanged during a dispatch cycle. For decentralized
strategies, nodes exchange load and generation information with one another resulting in 2m(m − 1)
data exchanges.
Forecast model inefficiencies can inject error into the computed dispatch vector due to generation
prediction inaccuracies. As the proposed distributed and existing decentralized strategies rely on real-
time generation data for dispatch and do not use forecast models over long time horizons, the error
introduced is close to null. Centralized online strategies can use short term (e.g. 3-hour) generation
models and offline strategies can use a-day-ahead prediction models. Errors introduced in these models
are provided in [11].
Global optimality of DG dispatch is an important consideration. The theoretical studies presented
Chapter 4. Distributed Generation Dispatch 81
earlier for the proposed technique guarantee optimality under certain non-restrictive conditions. Cen-
tralized online strategies may not guarantee an optimal solution due to the sheer size of the problem and
imposed time limitations (short optimization intervals). Moreover, distributed strategies such as that
outlined in reference [60] may not converge to optimality within the short optimization interval used for
real-time dispatch.
From the comparisons presented in the above, it can be concluded that the proposed dispatch strategy
provides a good balance amongst a variety of critical performance metrics.
4.4 Remarks
In this chapter, two dispatch proposals have been presented. The first solves economic dispatch so
that consumer demands can be balanced with available DG generation capacity at optimal costs. The
underlying physical constraints and discrete optimization variables are not accounted for in this proposal.
The second proposal is more generally applicable as this takes into account physical grid limitations while
balancing available generation with demands in a resilient manner. These proposals are shown to have
highly desirable convergence properties and scalability. In the next chapter, the proposals presented in
this chapter and the previous chapter are combined into a hierarchical algorithm in order to coordinate
both flexible demands and sustainable generation across both transmission and distribution networks.
Chapter 5
Hierarchical Optimization of
Distributed Generation and Demand
In this chapter, a hierarchical technique is proposed for the adaptive coordination of flexible demands
and DG generation spanning across both the transmission and distribution network levels so that the
overall operation of the power grid is sustainable, economical and resilient. This is a hybrid scheme that
taps onto the potential available in heterogeneous end nodes such as flexible consumers and distributed
energy sources (i.e. DGs and storage). In the proposed hierarchy, there are two main tiers and these are
referred to as transmission and distribution tiers.
The topmost level in the hierarchy is the transmission tier. Coordination in this tier takes place
amongst buses situated in the high-voltage transmission network level. This tier involves completely
decentralized optimization which is ideally suited to accommodate both bulk generation entities and
competitive energy markets stemming from deregulations. Traditionally, system operators managed
power generation operations in a central manner and this provided operators with unprecedented access
to these entities. With recent deregulations, central authorities will not have deep insights into the inner-
workings of private energy and service providers. Moreover, it will not be tractable to maintain details
about each and every transient private energy market participant. Furthermore, as competitive energy
markets are not yet fully embraced by all jurisdictions, coordinating surplus and deficits in generation
across various inter-ties and also accounting for unpredictable transmission line congestions will be chal-
lenging to accommodate centrally. The proposed decentralized process involves every EPU representing
a DN, characterized by local generation mix and flexible loads, coordinating with neighbouring EPUs
from other DNs to guide local participants according to global conditions spanning across the entire
transmission network. This allows for the seamless integration of a large number of heterogeneous enti-
ties such as flexible loads and DGs to meet system-wide goals. This approach leverages upon by convex
optimization based theoretical constructs such as alternating direction method of multipliers (ADMM)
and consensus protocols.
The second tier is referred to as the distribution tier as cooperation is driven by the EPU amongst
flexible consumers and DGs which are residing within the low-voltage DN managed by that EPU. In this
tier, the flexible loads adjust local demands and DGs dispatch power based on local generation capacities
to balance fluctuations in power consumption patterns while ensuring that the overall power injection
into the main grid computed in the transmission tier is maintained over the allocated optimization
82
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 83
interval. In the distribution tier, a central coordinating entity (which will typically be the EPU) transmits
general signals reflecting the current state of the DN which are then used by flexible end nodes in
that DN to make local adjustments to achieve the targetted power injection into the main grid in an
economical manner. In Chapters 3 and 4, distributed coordination techniques via population games have
been presented for both DR and DG dispatch across highly granular optimization intervals which can
effectively capture fluctuations in demand and generation levels. These are utilized in the distribution
tier for the coordination of local flexible consumers and DGs in this chapter.
This hierarchical approach allows for the decoupling and abstraction of various restrictions and
operating conditions associated with the DNs and competing power entities from one another. This allows
for the seamless plug-and-play integration of a wide variety of heterogeneous components into various
tiers of the hierarchy. Since only aggregate information is transferred between tiers and no individual data
is exchanged, privacy is enhanced. This method also allows for the safe injection of surplus generation in
DNs stemming from significant DG penetration in a manner that heeds voltage limitations in the DNs
while also ensuring that this surplus power contributes to reducing the deficit in sustainable generation
in other DNs. Moreover, flexible consumers are also leveraged to provision for dire conditions in which
generation capacity across the entire transmission network is simply not sufficient to sustainably and
economically meet demands in various DNs. This hierarchical technique promotes sustainable generation
and demand balance in the grid while also encouraging private sustainable generation competitors to
participate in energy markets. It can also be applied to hierarchical structures naturally present within
sub-systems contained within the power grid (e.g. microgrids or DNs).
5.1 System Model
The proposed hierarchical technique spans across transmission and distribution networks illustrated in
Fig 5.1. The buses in the transmission network are represented by EPUs managing individual DNs
Figure 5.1: System model for hierarchical topology.
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 84
(located in the transmission tier of Fig. 5.1). These buses can communicate with one another and are
equipped with agents that can make intelligent decisions. These bus agents communicate with only
neighbours in an iterative manner until a consensus on the optimal solution in the transmission tier is
reached. Unlike previous chapters, there are no central authorities broadcasting general information to
participating bus agents and therefore the optimization performed in the transmission tier is completely
decentralized involving peer-to-peer communications only. The transmission network typically has a
mesh topology consisting of lines that form cycles unlike the DNs which commonly have a tree structure.
This mesh network topology consisting of possibly multiple cycles will not be problematic in the proposed
solution as DC power flow formulation is used to derive the transmission tier algorithm. For economic
dispatch in high-voltage networks, the DC power flow formulation is typically utilized for frequency
control in the grid [78]. Each bus is associated with minimum and maximum values associated with
power injection which are computed based on the availability of local generation capacity and flexible
loads in the corresponding DNs. The optimization interval considered for the transmission tier is ten
minutes in length and therefore fluctuations in generation and dispatch can be closely captured by the
proposed algorithm.
Once coordination of power injection between bus agents in the transmission network is complete
over one optimization interval, the optimal setpoints for power injection is established at each bus and
these are used by EPUs to coordinate local flexible loads and DGs to achieve the precomputed power
injection setpoint. This distribution network level coordination (as illustrated in the distribution tier of
Fig. 5.1) occurs at a more granular level in order to ensure that variations in local consumer demands
and generation capacity flux are accounted for more accurately. Thus, the optimization interval at the
distribution tier is set to one minute. Coordination in DNs occur independently as these are decoupled
from one another by the decentralized optimization performed in the transmission tier.
5.1.1 Assumptions
Here, the assumptions made in the hierarchical algorithm are presented. In addition to the assumptions
made in Chapters 3 and 4, the following are further assumptions made for the hierarchical system which
now includes the transmission network:
1. There exists sufficient DG generation and flexible consumer demands to meet system goals;
2. Every bus in the transmission network is equipped with an intelligent agent (referred to as the bus
agent) which can also communicate with one another and if there is one DN attached to a bus in
the transmission network then the corresponding EPU will be the bus agent;
3. The optimization interval at the transmission tier is every 10 minutes;
4. The optimization interval at the distribution tier is every 1 minute;
5. Steady-state power grid conditions are considered in both tiers; and
6. Forecast of local demands and supply are available to every bus agent.
The first assumption is necessary for strong duality to hold and is also practical as ancillary generation
and storage systems will be available in standby mode in the event of deficit in power supply [106].
Assumption 2 coincides with the cyber-physical vision of the smart grid. Assumption 3 is reasonable
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 85
as economic dispatch over transmission networks take place over much larger timescales such as 24
hours [88]. The optimization interval being set to 10 minutes allows for the prevention of large error
margins introduced by forecast models over longer prediction horizons. Assumption 4 allows for the EPUs
to account for significant fluctuations in demand and supply within DNs. Assumption 5 is typically
the case in economic dispatch problem formulations [72]. The final assumption can be achieved by
DGs communicating to the EPU local generation capacities expected over 10 minute intervals. Demand
patterns are fairly consistent and are already available to the EPUs. Another option will be for individual
consumers to send local demands expected over the optimization intervals to data concentrators which
will then be aggregated and transmitted to the EPU.
5.2 Transmission Network Coordination
In Chapters 3 and 4, distributed algorithms consisting of a central entity that transmits general in-
formation about the state of the grid are used by participating agents to make local decisions. This
distributed solution structure with a central coordinating entity will not be practical in the transmission
network level due to the lack of homogeneity resulting from current trends in deregulations and the sheer
number of actuating nodes (i.e. thousands of flexible consumers and small/large scale DGs belonging to
hundreds of DNs) in the system. Hence, a completely decentralized algorithm is necessary for this. This
is facilitated by ADMM and consensus constructs from convex optimization and these are introduced
next.
5.2.1 Introduction to ADMM
Combining ADMM and consensus methods allow for completely decentralized decision-making that
involves communication with only adjacent neighbours. Suppose that the original formulation is a
convex optimization problem with the following general structure:
PA : minx∈Cx,y∈Cy
f(x) + g(y)
Ax+By = D (C1)A
where x and y are the optimization variables, f : R|x| → R and g : R|y| → R are convex functions, Cxand Cy are separable convex constraints associated with x and y, A ∈ Rc×|x| and B ∈ Rc×|y|, c is the
number of equality constraints in PA and D ∈ Rc is a vector of constants. The coupling constraint in
(C1)A defines a set of linear equality relationships between x and y. Without this constraint, PA will be
a completely separable and decomposable problem that can be solved independently by every agent in
the system. In order to overcome the difficulty caused by (C1)A, PA is relaxed to the following problem:
PdA : maxν
minx∈Cx,y∈Cy
Lρ(x, y, ν) = f(x) + g(y) + νT (Ax+By −D) +ρ
2||Ax+By −D||22
where ν ∈ Rc is a set of Lagrangian multipliers associated with the c equality constraints in (C1)A and
ρ > 0. Lρ(x, y, ν) is referred to as the augmented Lagrangian. The dual problem PdA can be solved
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 86
iteratively via ADMM as follows [121]:
xk+1 = argminx
Lρ(x, yk, νk)
yk+1 = argminy
Lρ(xk+1, y, νk)
νk+1 = νk + ρ(Axk+1 +Byk+1 −D)
where k is an index denoting the current updating iteration. The first two steps in ADMM minimize
the Lagrangian with respect to the primal variables x and y separately. This separability allows for
decentralized computation enabled by information exchanges. While minimizing Lρ with respect to x, y
and ν are held constant and are set to the values yk and νk computed in the previous iteration. When Lρ
is minimized with respect to y, x and ν are set to xk+1 computed immediately before and νk computed
in the last iteration. Finally, ν is updated using xk+1 and yk+1 computed in the current iteration and
the step-size is set to ρ which is the Lagrangian parameter in PdA. Information exchanges are necessary
during each one of these sequential update steps and these contain the precomputed parameters necessary
for the current update. Primal variables involve minimization and dual variables are associated with
maximization. Hence, these updates take place in alternating directions and thus referred to as ADMM.
This method will converge to optimality if strong duality holds and this is indeed the case due to
Assumption 1. As outlined in reference [119] linear convergence of distributed ADMM is guaranteed
under mild conditions. The convergence speed of the algorithm has been observed to be fast to attain
moderate accuracy which suffices for most applications [121]. Attaining finer accuracy via ADMM results
in much slower convergence. The degree of accuracy can be inferred by computing the residual r:
rk = ||Axk +Byk −D||2 (5.1)
where rk → 0 as k →∞ and optimality is achieved.
5.2.2 Problem Formulation
In order to take advantage of the decentralized solution structure and desireable convergence character-
istics enabled by ADMM, the optimization problem considered at the transmission level is transformed
using consensus-based techniques into a form to which ADMM can be applied. First, consider the
problem formulation presented in PT :
PT : minp
f(p) =∑n∈B
αn(pgn)2
pgn − pdn = −∑m∈Bn
bn,m(θn − θm) ∀ n ∈ B (C1)T
0 ≤ pgn ≤ cgn ∀ n ∈ B (C2)T
where pgn is the real power generation in bus n, pdn is the aggregate consumer demand from bus n,
Bn represents the set of buses adjacent to bus n and n ↔ m is a line adjacent to bus n (i.e. m is
a neighbour of n). This formulation is similar to the linear power flow problem presented in PFR of
Chapter 2. This consists of optimization variables that are real power generation p = [pg1 . . . pg|B|] and
the voltage angles θ = [θ1 . . . θ|B|]. The normalized voltage magnitudes in high-voltage transmission
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 87
networks are close to unity and conductances gn,m are much smaller than susceptances bn,m in lines
(n↔ m) ∈ L. These assumptions render reactive power flows negligible. It is evident that the objective
in PT and constraints in (C2)T are separable. However the power balance constraints in (C1)T consists
of variables from multiple buses (specifically neighbouring buses). This particular constraint prevents
the straightforward decomposition of PT into individual sub-problems.
5.2.3 Transformations to ADMM Form
Next, a set of transformations are applied to PT so that the resulting formulation is in a form similar
to PA to which ADMM can be directly applied. Constraint (C1)T represents the net power balance at
each bus. This power balance is governed by bus angles of neighbouring buses. In order to decouple the
net power balance equations, new variables are introduced. Each bus uses a subset of these variables
to keep track of its perspective of what the primary variables are of neighbouring buses. In particular,
the primary variables that are of main concern to each bus are the bus angles θ. For instance, consider
bus n and its neighbouring bus m where m ∈ Bn. If an agent residing in bus n is to compute the
local net power flow balance in (C1)T , it will require θm from all buses that are its neighbours. Bus n
can maintain a new variable θm,n which is the value of θm inferred from the perspective of bus n. At
optimality, this perspective variable θm,n must be equal to the original variable θm (i.e. θm,n = θm).
This equality enforces consensus between the new perspective variable and the original variable. Let all
of these perspective variables form the set y = [y1 . . . y|B|] and all the original variables form the set x
where each element in these sets is a tuple as follows:
yn = {pgn,n, θn,n, {θm,n| m ∈ Bn}} ∀ n ∈ B,
xn = {pgn, θn} ∀ n ∈ B
where yn contains all the perspective variables maintained by bus n and xn contains all of the original
variables local to bus n. In the set yn, new variables such as pgn,m where m ∈ Bn are not required as the
net power balance equation associated with a bus in (C1)T does not include real power generated from
any adjacent buses. The constraint sets Cx = {C1x ∪ . . . ∪ C
|B|x } and Cy = {C1
y ∪ . . . ∪ C|B|y } are defined as:
Cnx :
0 ≤ pgn ≤ cgn−∞ ≤ θn ≤ ∞
Cny :
pgn,n − pdn = −∑m∈Bn
bn,m(θn,n − θm,n)
It is important to note that the constraints sets Cnx and Cny are completely separable from one another as
these are based on local primary and perspective variables respectively. Equipped with these definitions,
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 88
the problem formulation in PT can be equivalently transformed into:
P ′T : minp
∑n∈B
fn(xn)
xn ∈ Cnx ∀ n ∈ B (C1)’T
yn ∈ Cny ∀ n ∈ B (C2)’T
xn = yn,m ∀ m ∈ {Bn, n}, n ∈ B (C3)’T
where yn,m is the perspective variable associated with the primary variable in bus m from the point of
view of bus n and the third constraint establishes consensus with the perspective and original variables.
This enforces the primary variables xn and the associated perspective variables in yn,m to be the same.
For example, θm is a primary variable of bus m and θm,n is the perspective variable of θm maintained
by bus n which is a neighbour of bus m. These variables must be the same (i.e. θm = θm,n) for the
feasibility of problem P ′T . P ′T is in essence equivalent to the original problem PT with the consensus
constraint in effect. The only difference is that more variables are maintained by each bus agent to allow
for separability. It is also important to note that P ′T is now of the same form as PA presented earlier in
this chapter to which ADMM can be applied. The dual of P ′T using the augmented Lagrangian is:
P ′Td
: maxν
minx∈Cx,y∈Cy
Lρ(x, y, ν) =∑n∈B
fn(xn) +∑n∈B
∑m∈{Bn,n}
νn,m(xn − yn,m) +ρ
2
∑n∈B
∑m∈{Bn,n}
(xn − yn,m)2
where ν ∈ R|y| with a slight overloading of notation where xn is referencing a single primary variable
not a set. Grouping terms that are associated with each bus, the Lagrangian can be decomposed into
sub-problems. When applying ADMM to solve P ′Td
iteratively, three sets of updates are made at every
iteration.
First, x (i.e. primary variables) are solved for by fixing yk and νk from the previous iteration.
By fixing y and ν into constant values, the above Lagrangian becomes completely separable. Ignoring
constant terms in Lρ(x, y, ν), each bus agent solves the following to obtain xk+1:
Pxn : xk+1 = [pgnk+1, θk+1
n ] = argmin(pgn,θn)∈Cnx
(αi + ρ)(pgn)2 + (νpn,nk − pgn,n
kρ)pgn+
+∑
m∈{Bn,n}
(ρθ2n + νθn,m
kθn − θn,mkθnρ)
where the optimization variables pgn and θn are highlighted in blue, νpn,nk is the Lagrangian variable
already computed from the last iteration k that is associated with the constraint pgn−pgn,n = 0, similarly
νθn,mk
is the Lagrangian variable also computed from the last iteration k which is associated with the con-
straint θn−θn,m = 0, pgn,nk is a y variable computed from the previous iteration k which is the perspective
from bus n the real power generation at the same bus (this variable is included to allow for separability)
and θkn,m is another y variable computed by the agent residing at bus m to obtain a perspective of θn
from bus m. Prior to solving the above optimization problem, every bus agent must gather precomputed
variables νθn,mk
and θkn,m from neighbours and locally computed variables pgn,nk, νpn,n
k and νθn,nk
in the
previous iteration. The transfer of these values is facilitated by communications established between
buses that are physical neighbours in the transmission network. Next, a set of updates are applied to
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 89
the y variables at each bus as follows:
Pyn : yk+1 = [pgn,n, θm,n| ∀ m ∈ {Bn, n}] = min(pgn,n,θm,n| ∀ m∈{Bn,n})∈Cny
(pgn,n)2ρ− (νpn,nk + pgn
k+1ρ)pgn,n
+∑
m∈{Bn,n}
(θ2m,nρ− (νθm,n
k+ θm
k+1ρ)θm,n)
where once again the optimization variables [pgn,n, θn,m| ∀ m ∈ {Bn, n}] are highlighted in blue. These
variables reflect bus n’s perspective of variables belonging to local and neighbouring buses that are
used in the coupling constraint (C1)T . The x and ν variables used as parameters in this optimiza-
tion problem that have been solved immediately prior to this update or in the previous iteration k
are pgnk+1, θm
k+1, νθm,nk
and νpn,nk where θm
k+1 and νθm,nk
are exchanged between neighbours. After
updating the y-variables, the final step in this ADMM iteration involves updating the dual variable ν
associated with the consensus constraints as follows:
νpn,nk+1 = νpn,n
k + ρ(pgnk+1 − pgn,n
k+1)
νθn,mk+1
= νθn,mk
+ ρ(θnk+1 − θn,mk+1) ∀ m ∈ {Bn, n}
which improves the Lagrangian variable ν via dual ascent. To compute these variable updates, x and
y variables that have been computed in the previous two steps are necessary in a manner similar to
the primary and perspective variable updates conducted earlier. For this, another set of information
exchanges are executed between neighbouring buses.
5.2.4 Summary of Decentralized Algorithm
Table 5.1 presents a summary of the decentralized algorithm used by bus agents to coordinate power
injections to achieve the optimal solution of PT .
In reference [74], ADMM and consensus protocol has been applied to a radial network (i.e. distribu-
tion network) to address a relaxed version of optimal power flow which is a semi-definite program (i.e.
convex optimization). This formulation is exact for the radial network under certain conditions [122].
The proposal presented in this chapter, on the other hand, applies ADMM and consensus to a mesh
network (i.e. transmission network) to achieve optimal linearized power flow. Economic dispatch in the
transmission level is typically based on the DC formulation and therefore additional complexities intro-
duced by the original OPF with non-linear constraints is not necessary. Furthermore, as the DC problem
is convex, the mesh structure of the transmission network will not be problematic in the application of
the proposed algorithm. If the problem is non-convex, then convergence to the optimal solution will not
be guaranteed. There are other proposals such as reference [75] that apply ADMM to the original OPF
without any relaxations. As the original OPF is non-convex, the proposed technique in reference [75]
has no strong guarantees of convergence.
5.2.5 Numerical Results
In this section, the transmission level coordination algorithm proposed via ADMM is implemented in
two transmission network systems using MATLAB. One is the WECC 3 generator 9 bus system and
the other is IEEE 10 generator 39 bus system which are illustrated in Figs. 5.2 and 5.3 respectively.
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 90
Decentralized Algorithm at the Transmission Tier
Initialization:
• Set xn ← 0, ym,n ← 0 ∀ m ∈ {Bn, n}, νn ← 0 andk ← 0
• Set cgn and pdn reflecting local DN’s generation and de-mands over the next 10 minutes
While r > ε:
1. Set k ← k + 1
2. Solve Pxn to obtain xkn using the initially computed
yk−1n and νk−1
n
3. Broadcast xkn to all neighbours m ∈ Bn4. Solve Py
n to obtain ykn using the initially computed xknand νk−1
n
5. Broadcast ykn to all neighbours m ∈ Bn
6. Compute νkn using previously computed νk−1n , xkn and
ykn
7. Broadcast νkn to all neighbours m ∈ Bn8. Compute r = ||x− y||2
Table 5.1: Decentralized algorithm for bus agent n in the transmission tier.
Although the WECC 3 generator 9 bus system is small in comparison to the IEEE 10 generator 39 bus
system, it provides means for comparison of the convergence properties of the ADMM algorithm in a
small versus large system. All parameters have been obtained using MATPOWER [124] and the base
MVA is 100. In these networks, every bus maintains its own set of variables and exchanges already
computed information amongst physically adjacent neighbours only.
Residual
The convergence properties of the ADMM algorithm for the WECC 3 generator 9 bus system is examined
first. Fig. 5.4 illustrates the residual (i.e. Eq. 5.1) computed over 1000 iterations in the WECC system
with the proposed algorithm in place. It is clear, that the residual stabilizes and approaches 0 within
500 iterations. Moreover, at iteration k = 114, the residual approaches 0 very fast and then resumes
to oscillate until k = 413. This steep descent of the algorithm reinforces the observation made earlier
regarding how the ADMM algorithm reaches moderate accuracy very quickly. The residual across the
iterations are not monotonously decreasing as the sub-problems constructed for each bus agent are not
differentiable due to local bus constraints.
Next, the convergence properties of the ADMM algorithm for the IEEE 10 generator 39 bus system is
depicted in Fig. 5.5. The initial value taken by each variable is 0. It is clear that initially, the algorithm
drives the system from a high residual to a residual of 5 × 10−3 very rapidly. However, as expected,
obtaining a fine accuracy in which the residual is very low requires a significant number of iterations.
Hence, if the system operator has some tolerance with respect to the accuracy of the solution, then it is
not necessary to fine-tune the solution.
The impact of the Lagrangian parameter ρ on the convergence speed of the algorithm is investigated
next. The smaller the step-size is, the more can the solution be fine-tuned but over larger number of
iterations. Fig. 5.6 illustrates the impact of varying ρ on the aggregate dispatch at k = 100 in the
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 91
Figure 5.2: WECC 3 bus line diagram, adapted from reference [123].
Figure 5.3: IEEE 39 bus line diagram, adapted from reference [27].
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 92
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
Residual versus Number of Iterations
Iterations
Resid
ual
Figure 5.4: Residual for WECC 9 bus system.
0 2000 4000 6000 8000 100000
0.005
0.01
0.015
0.02
0.025
Residual versus Number of Iterations
Iterations
Resid
ual
Figure 5.5: Residual for IEEE 39 bus system.
IEEE 39 bus system. The x-axis is based on a logarithmic scale and represents the step-size α. It is
evident that larger the step-size is, the faster is the solution approaching optimality. However, the speed
of convergence tapers off as α increases above 106. Additionally, the larger the value of ρ is, the less
fine-tuning can be applied to the resulting solution. Hence, the system operator will need to define ρ in
a manner that strikes a fine balance between speed and accuracy.
Comparison of Central and decentralized Methods
Next, the results of applying the proposed algorithm for the WECC 9 bus system depicted in Fig. 5.2
is listed in Table 5.2. In this table, the decentralized ADMM algorithm is compared with the values
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 93
103
104
105
106
107
0
0.5
1
1.5
2
2.5
3
Impact of α on Convergence Speed of ADMM
Step Size (α)
Ag
gre
gate
So
luti
on
Figure 5.6: Impact of Lagrangian step-size on ADMM convergence.
obtained from applying a central quadratic solver in MATLAB to the DC formulation. The central
solution is the exact solution whereas the decentralized solution is obtained by terminating the algorithm
when the residual is less than 9 × 10−4. The power demand at each bus is randomly assigned. There
are 18 primary variables in this particular example and when the perspective variables are included, the
optimization space becomes much larger. It can be observed that the optimal generation at each bus and
the bus angles obtained via the decentralized algorithm is very close the exact solution obtained using
a central solver. Hence, this illustrates that if the system operator has some tolerance to the accuracy
of the solution, then the proposed algorithm is suitable for decentralized computation within practical
number of iterations.
Bus Bus Bus Generation Bus Generation Bus Angle Bus AngleNumber Demand (Central) (decentralized) (Central) (decentralized)
1 0 38.3333 38.3467 0.0000 02 20 38.3333 38.3617 -0.6447 -0.64383 10 38.3333 38.3621 0.3157 0.31674 30 0.0000 0 -2.2080 -2.20875 10 0.0000 0 -2.5295 -2.53026 15 0.0000 0 -1.3446 -1.34497 10 0.0000 0 -2.0306 -2.03148 10 0.0000 0 -1.7906 -1.79119 10 0.0000 0 -2.6350 -2.6358
Table 5.2: Comparison of results from central versus decentralized algorithms WECC 9 bus system.
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 94
The results obtained for the IEEE 39 bus system illustrated in Fig. 5.3 are listed in Table 5.3.
Once again, the problem is solved using a central quadratic solver and these results represent the exact
solution. The decentralized ADMM algorithm is also applied to the system. There are 78 primary
variables maintained in the system in addition to the perspective variables amongst which consensus
must be reached. The values listed in Table 5.3 are associated with a residual of 2× 10−8. These values
are fairly close to the exact solution obtained via a central solver. These results demonstrate the efficacy
of the proposed decentralized solution for the transmission tier.
Bus Bus Bus Generation Bus Generation Bus Angle Bus AngleNumber Demand (Central) (decentralized) (Central) (decentralized)
1 0 0.0000 0 0.0000 02 0 0.0000 0 0.0691 0.06943 1 0.0000 0 -0.0431 -0.04304 1 0.0000 0 -0.1827 -0.18375 4 0.0000 0 -0.2557 -0.25736 5 0.0000 0 -0.2556 -0.25727 2 0.0000 0 -0.2749 -0.27658 4 0.0000 0 -0.2753 -0.27699 4 0.0000 0 -0.1963 -0.197010 0 0.0000 0 -0.2433 -0.244911 3 0.0000 0 -0.2584 -0.260112 2 0.0000 0 -0.2945 -0.296013 5 0.0000 0 -0.2434 -0.244914 0 0.0000 0 -0.1808 -0.181815 2 0.0000 0 -0.0432 -0.043216 2 0.0000 0 0.0355 0.035917 2 0.0000 0 -0.0066 -0.006118 4 0.0000 0 -0.0409 -0.040619 1 0.0000 0 0.1730 0.173120 4 0.0000 0 0.1799 0.180021 2 0.0000 0 0.1210 0.121422 0 0.0000 0 0.2377 0.238123 1 0.0000 0 0.2455 0.245824 4 0.0000 0 0.0455 0.046025 2 0.0000 0 0.1340 0.134626 0 0.0000 0 0.0701 0.071027 2 0.0000 0 0.0188 0.019628 3 0.0000 0 0.0886 0.089929 1 0.0000 0 0.1402 0.141630 4 8.5000 8.5025 0.1505 0.150931 1 8.5000 8.5021 -0.0681 -0.069732 5 8.5000 8.5007 -0.1733 -0.174933 1 8.5000 8.4906 0.2797 0.279834 4 8.5000 8.4897 0.2611 0.261035 1 8.5000 8.4903 0.3450 0.345236 1 8.5000 8.4899 0.4496 0.449637 0 8.5000 8.5003 0.3313 0.331938 3 8.5000 8.4946 0.2263 0.227639 4 8.5000 8.5136 -0.0418 -0.0420
Table 5.3: Comparison of results from central versus decentralized algorithms IEEE 39 bus system.
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 95
5.2.6 Comparison with State-of-the-art Decentralized Techniques
Generally, the performance of ADMM is shown to be linear with the size of the system under some
mild conditions [74, 119]. Other decentralized algorithms used commonly in the state-of-the-art include
average consensus and dual decomposition via sub-gradient methods. The convergence rate of average
consensus is dependent on the second smallest eigenvalue of the Laplacian matrix representing the
underlying network topology. In the worst case, this is O(n3) [120]. Convergence of dual decomposition
via sub-gradient method depends on the step-size used to update the dual variables. In addition to
performance, another advantage of ADMM is that it naturally applies to the underlying decentralized
structure of the optimization problem at hand [106].
5.3 Distribution Network Level Coordination
Next, the coordination of flexible demands and DGs in the distribution tier is presented. It is important
to note that curtailment of flexible demand can be considered to be positive generation. At the beginning
of each optimization interval, the EPU coordinates the dispatch of DGs to achieve the pgn computed in the
transmission tier according to the algorithm presented in Table 4.2 in Chapter 4. In this computation,
the offset between demand forecast used at the transmission level denoted as∑d∈D p
′d and current
demands∑d∈D pd must also be accounted for. When the overall dispatch poverallg at the end of the DG
coordination is not sufficient to meet the targeted pgn, then the EPU coordinates demand curtailment
in flexible consumers in order to meet the targeted power generation in accordance to the algorithm
presented in Chapter 3 in Table 3.3. For this, the setpoint for demand curtailment is set to:
SDR = pgn − poverallg +∑d∈D
pd −∑d∈D
p′d (5.2)
This distribution tier coordination in summarized in Table 5.4. Separation between DR and DG
dispatch is necessary as the demand curtailment levels will not necessarily coincide with the generation
dispatch levels. Moreover, demand curtailment should be a last resort option as consumers will naturally
prefer to operate appliances as usual. All participating entities (i.e. flexible consumers or DGs) will
respond to the signals transmitted by the EPU as summarized in Tables 3.3 and 4.2 in Chapters 3 and 4.
The DR coordination curtails loads in accordance to local operating conditions and consumer comfort
levels. The DG coordination takes into account voltage rise considerations which are imperative in DNs.
This combined approach used to achieve the targeted power injection computed in the transmission tier
within the short optimization interval of one minute is feasible as the convergence of these algorithms
is exponential when projection revisions are used (as demonstrated in Chapter 4). Hence, convergence
of both of these coordination algorithms operating sequentially can be achieved within the one minute
optimization interval allocated to the distribution tier level optimization.
5.3.1 Resilience
In the coordination of vast number of cyber-enabled heterogeneous entities that rely upon communica-
tion broadcasts to make local revisions, resilience is an important factor. As demonstrated in Chapters
3 and 4, the proposed distributed DR and DG dispatch algorithms are associated with strong resilience
Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 96
Distributed Coordination by the EPU at the Distri-bution Tier
For every optimization interval:
1. Coordinate DGs residing in the local DN using thealgorithm in 4.2 where
∑d∈D pd is replaced with∑
d∈D pd −∑
d∈D p′d + pgn until |xk − xk−1| < ε
2. Coordinate DR participants residing in the local DNusing the algorithm listed in 3.3 where SDR is set tothe value obtained from Eq. 5.2. During every revision,perform voltage feasibility check
Table 5.4: Decentralized algorithm for bus agent n at the transmission tier.
properties. Hence, in the event of perturbations, other agents in the system will be able to infer this
implicitly through broadcast signals and act appropriately. As these algorithms are utilized for distri-
bution tier coordination in this chapter, all properties associated with resilience are also inherited in
the distribution tier coordination. Thus, resilience is built into the core of the design of the proposed
hierarchical optimization algorithm.
5.4 Remarks
The hierarchical technique presented in this chapter allows for the decentralized and distributed co-
ordination of heterogeneous power grid components. Superior convergence characteristics obtained by
applying population game theory constructs are tapped upon to enable the seamless coexistence of flex-
ible loads and DGs with extremely different local operational conditions using a central coordinating
entity like the EPU in the distribution tier. Moreover, resilience innate in these algorithms is inherited
in the distribution tier level. In order to abstract operational details of individual DNs, a completely de-
centralized algorithm with fast convergence properties obtained by leveraging on ADMM and consensus
theoretical constructs is proposed to manage the transmission network tier. This decentralized approach
is ideal for managing independent entities such as IPP, bulk generation and power supply from DNs in
an effective manner.
Chapter 6
Conclusions
Emerging trends in the power grid are introducing many advances and changes to the traditional energy
infrastructure and operational processes. Power components are no longer passive as these can now
communicate, learn, monitor and make intelligent decisions. This integration of the cyber domain and
intelligence with the physical grid gives rise to self-healing, resilient and adaptive decision-making ability
by many actuating grid components. The evolution of the power grid into a smart grid also, inevitably,
introduces many challenges due to the high penetration of fluctuating generation systems, deregulations,
cyber-security and so on. In this thesis, it is demonstrated that these challenges can be effectively
overcome by tapping onto the connectivity and intelligence enabled by the cyber-physical smart grid.
Self-adaptive responses by individual power components can be designed by bridging mathematical
theoretical constructs with the underlying physics of the power grid.
6.1 Summary
As such, a basic overview of the evolving landscape of the power grid is presented first in this thesis.
Here, the advantages and challenges due to these changes are introduced and these set the premise for
the work presented in this thesis. Then, various optimization problems encountered in the power grid
along with the method for gauging the difficulty of solving these are also introduced. This exposition
serves to highlight the need for alternative methods to overcome difficulties caused by non-convexity
and in-scalability of the traditional optimization formulations. The general theme in these problems
is the presence of thousands of actuating nodes each with individual operating limitations which are
bound together by a global constraint. These nodes must be directed in an effective and economical
manner so that a particular goal (i.e. minimize cost, carbon footprint, etc.) can be achieved in the
system. Using traditional approaches such as central optimization for directing these nodes in an optimal
manner is mathematically intractable and imposes extensive resource overheads. Hence, distributed and
decentralized approaches are alternative solutions whereby individual nodes are presented with some
general information about the global state of the system at periodic intervals which are then used by these
nodes to make individual decisions based on local operating conditions. This distributed/decentralized
solution structure capitalizes on the individual monitoring, communication and actuation capabilities of
active power components. Moreover, recent developments in data aggregation and wireless broadcast
capabilities are also levied to keep individual nodes informed about the current state of the system.
97
Chapter 6. Conclusions 98
This approach of offloading decision-making to a large number of small-scale actuating devices es-
sentially parallelizes computations and taps onto the vast potential of the cyber-physical amalgamation
in the power grid. However, in order to ensure that these individual decision-making entities converge
to the optimal operating state, it is necessary to demonstrate both theoretically and practically that the
proposed distributed algorithms are indeed associated with desirable convergence traits. Proposals pre-
sented in this thesis utilize convex optimization and population game theoretic constructs to construct
distributed and decentralized algorithms to coordinate various power grid components such as demand
curtailment of flexible consumers and DG dispatch. Dynamics induced by these distributed decisions are
proven to have superior convergence properties (asymptotic and exponential) to optimality regardless of
the initial state or the significant number of participants in the system. Moreover, it is demonstrated
that the system is robust and resilient to perturbations that can be caused by fluctuating local operating
conditions of the grid and/or cyber attacks. These results are verified in practical MATLAB simulations
with realistic models of consumer demands and DG generation capacities. In the culmination of the
thesis, a hierarchical solution is presented for the coordination of heterogeneous components such as
flexible consumers and DGs across multiple tiers in the power grid.
Thus, in this thesis, principles of abstraction and decoupling are applied to non-intrusively and
seamlessly integrate diverse components in the power grid to promote economical and sustainable power
grid operations.
6.2 Contributions
In this thesis, novel distributed and decentralized algorithms have been presented to allow individual
actuating components with local operating constraints and global operational constraints to make their
own actuation decisions based on external information obtained through either a central coordinating
entity or peer-to-peer information sharing. These proposals are highly scalable with strong convergence
characteristics. Hence, in this thesis, the seamless integration of a large number (thousands) of hetero-
geneous power components to achieve the common goal of economical and sustainable grid operations
in real-time has been investigated. Main contributions of this thesis are summarized in the following:
1. Novel approach to DR is presented that is: 1) Distributed: computational processing is offloaded
to cyber-physical DR agents located at consumer premises; 2) Consumer-centric: end-users
configure load operation preferences on their respective DR agents based on comfort requirements;
3) Real-time: the EPU broadcasts unidirectional communication signals employed by DR agents
to elicit load shedding decisions based on current local conditions and these decisions rapidly
converge to optimal aggregate peak reduction; 4) Resilient: it is possible to rapidly recover from
system perturbations when a subset of DR agents are corrupted; 5) Scalable: better convergence
is achieved with a larger number of participants.
2. Novel approach to dispatch of a large number of distributed energy sources is presented that is: 1)
Distributed: Computational efforts are offloaded to local cyber-physical DG agents that compute
their own dispatch via lightweight communications from the utility; 2) Real-time: DGs use
current local generation conditions for dispatch instead of long-term forecast models; 3) Scalable:
DG addition has little impact on the complexity and convergence properties of the algorithm; 4)
Efficient: Only unidirectional utility broadcast communication is required; signals transmitted
Chapter 6. Conclusions 99
by the utility enable rapid dispatch convergence. Moreover, the proposed mechanism does not
require communication of individual DG state information, but merely an aggregate providing
inherent privacy and resilience to denial-of-service attacks; 5) Resilient: As the system dynamic
is exponentially stable, the DG agents are able to recover from perturbations rapidly and restore
the optimal equilibrium.
3. Hierarchical approach combining DR and dispatch is presented that is: 1) Decentralized at
the transmission network tier and distributed at the DN level; 2) Real-time to accommodate
fluctuations in demand and generation; 3) Scalable: large number of nodes across the power grid
can be coordinated in real-time; 4) Accommodates heterogeneous devices and thereby supports
diversity in actuating devices; 5) Resilient: As algorithms with strong resilience characteristics
are deployed in the distribution tier, these properties are inherited by the proposed hierarchical
algorithm.
4. All of the convergence characteristics have been verified via theoretical studies and realistic simu-
lations using practical models and real data.
These algorithms are practical and can be applied by the grid operators to enable the sustainable, eco-
nomical and resilient grid operations. Moreover, these solutions effectively overcome many challenges
associated with high penetration of DGs, large power consuming devices, deregulations and existing vul-
nerabilities in the communication protocols. These solutions have been obtained by applying theoretical
constructs in optimization and game theory.
6.3 Future Directions
This thesis has explored how decentralized and distributed methods can be designed to enable intelligent
actuating nodes to respond to various changes in the power grid in an optimal manner. These efforts
have specifically focussed on flexible demands and sustainable generation sources. These applications
demonstrate the power of self-healing and adaptive actions of power grid components which have been
mainly enabled by the cyber-physical smart grid. This automated distributed decision-making overcomes
many challenges due to resource overheads, security and tractability. As such, the following are some
future extensions to this work:
• Voltage control study should be extended to unbalanced three phase radial DN for completeness.
• Microgrids and DNs can be leveraged in grid-connected mode to promote resiliency in the power
grid. These systems can be configured to automatically come online based on the overall grid
conditions and have the potential of being more economical than traditional ancillary services.
• Investigate how convex optimization techniques can be applied for optimal primary control in
microgrid settings. Control in microgrid components has been traditionally challenging due to the
extremely granular time margins available for achieving stability in the system (in the range of
nanoseconds). Using communication mechanisms such as large-scale broadcasts result in latencies
that are not acceptable in these time-scales. Hence, a completely decentralized solution with
minimal communications is necessary to address primary control in the microgrid level.
Chapter 6. Conclusions 100
• With self-actuating devices, it is necessary to ensure that safe operating limits and additional
protection mechanisms are integrated into these actuating nodes in order to prevent malicious
cyber attacks that can undermine the underlying stability of the system.
• Validations of proposed techniques in this thesis have been implemented using realistic models
and parameters in MATLAB. In order to ensure that these have acceptable performance in real
systems, advanced simulation systems such as Opal-RT or RTDS should be utilized. These can
also incorporate advance models to take into account transients and changes resulting from the
underlying physical attributes of the grid.
• In the hierarchical technique presented in this thesis, DC problem formulation has been used which
ignores the impact of reactive power flow. Extending this work to examine the effects of reactive
power flow in the transmission network on line loadability and other effects will provide interesting
insights into how system protection devices can be deployed to protect the grid.
• Investigate the impact when the system does not converge within the allocated optimization interval
(i.e. parameters change prior to convergence) via multi-period optimal power flow with a receding
horizon.
Hence, this particular area of research in the smart grid is rich and rewarding. There are many research
opportunities that can serve to enhance the security, sustainability and economical operations of the
power grid.
Appendix: List of Acronyms
DN Distribution Network EGT Evolutionary Game Theory
AGC Automatic Generator Control OPF Optimal Power Flow
DG Distributed Generator ED Economic Dispatch
MPP Maximum Power Point VR Voltage Regulation
IoT Internet of Things FR Frequency Regulation
DR Demand Response SOC State-of-Charge
DLC Direct Load Control KKT Karush Kuhn Tucker
EPU Electric Power Utility NS Nash Stationarity
TOU Time of Use NE Nash Equilibrium
HEMS Home Energy Management ESS Evolutionary Stable State
Systems AC Alternating Current
NIST National Institute of Standards NP Non-deterministic
and Technology Polynomial time
HVAC Heating, Ventilating PC Pairwise Comparison
and Air-Conditioning SLLN Strong Law of
ISO Independent System Operator Large Numbers
IPP Independent Power Plant NC Negative Correlation
ADMM Alternative Direction MIMO Multiple Input
Method of Multipliers Multiple Output
SG Sub-Gradient RD Replicator Dynamic
IEEE Institute of Electrical and WECC Western Electricity
Electronics Engineers Coordinating Council
101
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